author | wenzelm |
Sat, 05 Jan 2019 17:24:33 +0100 | |
changeset 69597 | ff784d5a5bfb |
parent 67565 | e13378b304dd |
child 69605 | a96320074298 |
permissions | -rw-r--r-- |
30439 | 1 |
(* Title : HOL/Decision_Procs/Dense_Linear_Order.thy |
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Plain, Main form meeting points in import hierarchy
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Author : Amine Chaieb, TU Muenchen |
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Plain, Main form meeting points in import hierarchy
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*) |
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section \<open>Dense linear order without endpoints |
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and a quantifier elimination procedure in Ferrante and Rackoff style\<close> |
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Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
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theory Dense_Linear_Order |
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imports Main |
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begin |
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Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
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ML_file "langford_data.ML" |
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ML_file "ferrante_rackoff_data.ML" |
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||
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context linorder |
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begin |
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Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
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lemma less_not_permute[no_atp]: "\<not> (x < y \<and> y < x)" |
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by (simp add: not_less linear) |
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lemma gather_simps[no_atp]: |
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"(\<exists>x. (\<forall>y \<in> L. y < x) \<and> (\<forall>y \<in> U. x < y) \<and> x < u \<and> P x) \<longleftrightarrow> |
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(\<exists>x. (\<forall>y \<in> L. y < x) \<and> (\<forall>y \<in> (insert u U). x < y) \<and> P x)" |
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"(\<exists>x. (\<forall>y \<in> L. y < x) \<and> (\<forall>y \<in> U. x < y) \<and> l < x \<and> P x) \<longleftrightarrow> |
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(\<exists>x. (\<forall>y \<in> (insert l L). y < x) \<and> (\<forall>y \<in> U. x < y) \<and> P x)" |
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"(\<exists>x. (\<forall>y \<in> L. y < x) \<and> (\<forall>y \<in> U. x < y) \<and> x < u) \<longleftrightarrow> |
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(\<exists>x. (\<forall>y \<in> L. y < x) \<and> (\<forall>y \<in> (insert u U). x < y))" |
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"(\<exists>x. (\<forall>y \<in> L. y < x) \<and> (\<forall>y \<in> U. x < y) \<and> l < x) \<longleftrightarrow> |
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(\<exists>x. (\<forall>y \<in> (insert l L). y < x) \<and> (\<forall>y \<in> U. x < y))" |
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by auto |
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lemma gather_start [no_atp]: "(\<exists>x. P x) \<equiv> (\<exists>x. (\<forall>y \<in> {}. y < x) \<and> (\<forall>y\<in> {}. x < y) \<and> P x)" |
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by simp |
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text\<open>Theorems for \<open>\<exists>z. \<forall>x. x < z \<longrightarrow> (P x \<longleftrightarrow> P\<^sub>-\<^sub>\<infinity>)\<close>\<close> |
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lemma minf_lt[no_atp]: "\<exists>z . \<forall>x. x < z \<longrightarrow> (x < t \<longleftrightarrow> True)" by auto |
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lemma minf_gt[no_atp]: "\<exists>z . \<forall>x. x < z \<longrightarrow> (t < x \<longleftrightarrow> False)" |
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by (simp add: not_less) (rule exI[where x="t"], auto simp add: less_le) |
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lemma minf_le[no_atp]: "\<exists>z. \<forall>x. x < z \<longrightarrow> (x \<le> t \<longleftrightarrow> True)" by (auto simp add: less_le) |
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lemma minf_ge[no_atp]: "\<exists>z. \<forall>x. x < z \<longrightarrow> (t \<le> x \<longleftrightarrow> False)" |
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by (auto simp add: less_le not_less not_le) |
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lemma minf_eq[no_atp]: "\<exists>z. \<forall>x. x < z \<longrightarrow> (x = t \<longleftrightarrow> False)" by auto |
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now use "Named_Thms" for "noatp", and renamed "noatp" to "no_atp"
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lemma minf_neq[no_atp]: "\<exists>z. \<forall>x. x < z \<longrightarrow> (x \<noteq> t \<longleftrightarrow> True)" by auto |
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now use "Named_Thms" for "noatp", and renamed "noatp" to "no_atp"
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lemma minf_P[no_atp]: "\<exists>z. \<forall>x. x < z \<longrightarrow> (P \<longleftrightarrow> P)" by blast |
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text\<open>Theorems for \<open>\<exists>z. \<forall>x. x < z \<longrightarrow> (P x \<longleftrightarrow> P\<^sub>+\<^sub>\<infinity>)\<close>\<close> |
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lemma pinf_gt[no_atp]: "\<exists>z. \<forall>x. z < x \<longrightarrow> (t < x \<longleftrightarrow> True)" by auto |
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lemma pinf_lt[no_atp]: "\<exists>z. \<forall>x. z < x \<longrightarrow> (x < t \<longleftrightarrow> False)" |
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by (simp add: not_less) (rule exI[where x="t"], auto simp add: less_le) |
34cb0b457dcc
Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
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lemma pinf_ge[no_atp]: "\<exists>z. \<forall>x. z < x \<longrightarrow> (t \<le> x \<longleftrightarrow> True)" by (auto simp add: less_le) |
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now use "Named_Thms" for "noatp", and renamed "noatp" to "no_atp"
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lemma pinf_le[no_atp]: "\<exists>z. \<forall>x. z < x \<longrightarrow> (x \<le> t \<longleftrightarrow> False)" |
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by (auto simp add: less_le not_less not_le) |
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now use "Named_Thms" for "noatp", and renamed "noatp" to "no_atp"
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lemma pinf_eq[no_atp]: "\<exists>z. \<forall>x. z < x \<longrightarrow> (x = t \<longleftrightarrow> False)" by auto |
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now use "Named_Thms" for "noatp", and renamed "noatp" to "no_atp"
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lemma pinf_neq[no_atp]: "\<exists>z. \<forall>x. z < x \<longrightarrow> (x \<noteq> t \<longleftrightarrow> True)" by auto |
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now use "Named_Thms" for "noatp", and renamed "noatp" to "no_atp"
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lemma pinf_P[no_atp]: "\<exists>z. \<forall>x. z < x \<longrightarrow> (P \<longleftrightarrow> P)" by blast |
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lemma nmi_lt[no_atp]: "t \<in> U \<Longrightarrow> \<forall>x. \<not>True \<and> x < t \<longrightarrow> (\<exists>u\<in> U. u \<le> x)" by auto |
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lemma nmi_gt[no_atp]: "t \<in> U \<Longrightarrow> \<forall>x. \<not>False \<and> t < x \<longrightarrow> (\<exists>u\<in> U. u \<le> x)" |
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Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
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by (auto simp add: le_less) |
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lemma nmi_le[no_atp]: "t \<in> U \<Longrightarrow> \<forall>x. \<not>True \<and> x\<le> t \<longrightarrow> (\<exists>u\<in> U. u \<le> x)" by auto |
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lemma nmi_ge[no_atp]: "t \<in> U \<Longrightarrow> \<forall>x. \<not>False \<and> t\<le> x \<longrightarrow> (\<exists>u\<in> U. u \<le> x)" by auto |
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lemma nmi_eq[no_atp]: "t \<in> U \<Longrightarrow> \<forall>x. \<not>False \<and> x = t \<longrightarrow> (\<exists>u\<in> U. u \<le> x)" by auto |
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lemma nmi_neq[no_atp]: "t \<in> U \<Longrightarrow>\<forall>x. \<not>True \<and> x \<noteq> t \<longrightarrow> (\<exists>u\<in> U. u \<le> x)" by auto |
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lemma nmi_P[no_atp]: "\<forall>x. ~P \<and> P \<longrightarrow> (\<exists>u\<in> U. u \<le> x)" by auto |
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lemma nmi_conj[no_atp]: "\<lbrakk>\<forall>x. \<not>P1' \<and> P1 x \<longrightarrow> (\<exists>u\<in> U. u \<le> x) ; |
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\<forall>x. \<not>P2' \<and> P2 x \<longrightarrow> (\<exists>u\<in> U. u \<le> x)\<rbrakk> \<Longrightarrow> |
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\<forall>x. \<not>(P1' \<and> P2') \<and> (P1 x \<and> P2 x) \<longrightarrow> (\<exists>u\<in> U. u \<le> x)" by auto |
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lemma nmi_disj[no_atp]: "\<lbrakk>\<forall>x. \<not>P1' \<and> P1 x \<longrightarrow> (\<exists>u\<in> U. u \<le> x) ; |
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\<forall>x. \<not>P2' \<and> P2 x \<longrightarrow> (\<exists>u\<in> U. u \<le> x)\<rbrakk> \<Longrightarrow> |
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\<forall>x. \<not>(P1' \<or> P2') \<and> (P1 x \<or> P2 x) \<longrightarrow> (\<exists>u\<in> U. u \<le> x)" by auto |
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lemma npi_lt[no_atp]: "t \<in> U \<Longrightarrow> \<forall>x. \<not>False \<and> x < t \<longrightarrow> (\<exists>u\<in> U. x \<le> u)" by (auto simp add: le_less) |
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lemma npi_gt[no_atp]: "t \<in> U \<Longrightarrow> \<forall>x. \<not>True \<and> t < x \<longrightarrow> (\<exists>u\<in> U. x \<le> u)" by auto |
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lemma npi_le[no_atp]: "t \<in> U \<Longrightarrow> \<forall>x. \<not>False \<and> x \<le> t \<longrightarrow> (\<exists>u\<in> U. x \<le> u)" by auto |
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lemma npi_ge[no_atp]: "t \<in> U \<Longrightarrow> \<forall>x. \<not>True \<and> t \<le> x \<longrightarrow> (\<exists>u\<in> U. x \<le> u)" by auto |
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lemma npi_eq[no_atp]: "t \<in> U \<Longrightarrow> \<forall>x. \<not>False \<and> x = t \<longrightarrow> (\<exists>u\<in> U. x \<le> u)" by auto |
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lemma npi_neq[no_atp]: "t \<in> U \<Longrightarrow> \<forall>x. \<not>True \<and> x \<noteq> t \<longrightarrow> (\<exists>u\<in> U. x \<le> u )" by auto |
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lemma npi_P[no_atp]: "\<forall>x. ~P \<and> P \<longrightarrow> (\<exists>u\<in> U. x \<le> u)" by auto |
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lemma npi_conj[no_atp]: "\<lbrakk>\<forall>x. \<not>P1' \<and> P1 x \<longrightarrow> (\<exists>u\<in> U. x \<le> u) ; \<forall>x. \<not>P2' \<and> P2 x \<longrightarrow> (\<exists>u\<in> U. x \<le> u)\<rbrakk> |
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\<Longrightarrow> \<forall>x. \<not>(P1' \<and> P2') \<and> (P1 x \<and> P2 x) \<longrightarrow> (\<exists>u\<in> U. x \<le> u)" by auto |
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lemma npi_disj[no_atp]: "\<lbrakk>\<forall>x. \<not>P1' \<and> P1 x \<longrightarrow> (\<exists>u\<in> U. x \<le> u) ; \<forall>x. \<not>P2' \<and> P2 x \<longrightarrow> (\<exists>u\<in> U. x \<le> u)\<rbrakk> |
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\<Longrightarrow> \<forall>x. \<not>(P1' \<or> P2') \<and> (P1 x \<or> P2 x) \<longrightarrow> (\<exists>u\<in> U. x \<le> u)" by auto |
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Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
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diff
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85 |
|
49519 | 86 |
lemma lin_dense_lt[no_atp]: |
87 |
"t \<in> U \<Longrightarrow> |
|
60539 | 88 |
\<forall>x l u. (\<forall>t. l < t \<and> t < u \<longrightarrow> t \<notin> U) \<and> l < x \<and> x < u \<and> x < t \<longrightarrow> (\<forall>y. l < y \<and> y < u \<longrightarrow> y < t)" |
89 |
proof clarsimp |
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49519 | 90 |
fix x l u y |
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assume tU: "t \<in> U" |
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and noU: "\<forall>t. l < t \<and> t < u \<longrightarrow> t \<notin> U" |
|
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and lx: "l < x" |
|
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and xu: "x < u" |
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and px: "x < t" |
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and ly: "l < y" |
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and yu: "y < u" |
|
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from tU noU ly yu have tny: "t \<noteq> y" by auto |
|
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have False if H: "t < y" |
|
100 |
proof - |
|
101 |
from less_trans[OF lx px] less_trans[OF H yu] have "l < t \<and> t < u" |
|
102 |
by simp |
|
103 |
with tU noU show ?thesis |
|
104 |
by auto |
|
105 |
qed |
|
106 |
then have "\<not> t < y" |
|
107 |
by auto |
|
108 |
then have "y \<le> t" |
|
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by (simp add: not_less) |
|
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then show "y < t" |
|
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using tny by (simp add: less_le) |
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112 |
qed |
34cb0b457dcc
Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff
changeset
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113 |
|
49519 | 114 |
lemma lin_dense_gt[no_atp]: |
115 |
"t \<in> U \<Longrightarrow> |
|
60539 | 116 |
\<forall>x l u. (\<forall>t. l < t \<and> t < u \<longrightarrow> t \<notin> U) \<and> l < x \<and> x < u \<and> t < x \<longrightarrow> (\<forall>y. l < y \<and> y < u \<longrightarrow> t < y)" |
117 |
proof clarsimp |
|
49519 | 118 |
fix x l u y |
60539 | 119 |
assume tU: "t \<in> U" |
120 |
and noU: "\<forall>t. l < t \<and> t < u \<longrightarrow> t \<notin> U" |
|
121 |
and lx: "l < x" |
|
122 |
and xu: "x < u" |
|
123 |
and px: "t < x" |
|
124 |
and ly: "l < y" |
|
125 |
and yu: "y < u" |
|
126 |
from tU noU ly yu have tny: "t \<noteq> y" by auto |
|
127 |
have False if H: "y < t" |
|
128 |
proof - |
|
129 |
from less_trans[OF ly H] less_trans[OF px xu] have "l < t \<and> t < u" |
|
130 |
by simp |
|
131 |
with tU noU show ?thesis |
|
132 |
by auto |
|
133 |
qed |
|
134 |
then have "\<not> y < t" |
|
135 |
by auto |
|
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then have "t \<le> y" |
|
137 |
by (auto simp add: not_less) |
|
138 |
then show "t < y" |
|
139 |
using tny by (simp add: less_le) |
|
49519 | 140 |
qed |
141 |
||
142 |
lemma lin_dense_le[no_atp]: |
|
143 |
"t \<in> U \<Longrightarrow> |
|
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\<forall>x l u. (\<forall>t. l < t \<and> t < u \<longrightarrow> t \<notin> U) \<and> l < x \<and> x < u \<and> x \<le> t \<longrightarrow> (\<forall>y. l < y \<and> y < u \<longrightarrow> y \<le> t)" |
145 |
proof clarsimp |
|
26161
34cb0b457dcc
Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff
changeset
|
146 |
fix x l u y |
60539 | 147 |
assume tU: "t \<in> U" |
148 |
and noU: "\<forall>t. l < t \<and> t < u \<longrightarrow> t \<notin> U" |
|
149 |
and lx: "l < x" |
|
150 |
and xu: "x < u" |
|
151 |
and px: "x \<le> t" |
|
152 |
and ly: "l < y" |
|
153 |
and yu: "y < u" |
|
154 |
from tU noU ly yu have tny: "t \<noteq> y" by auto |
|
155 |
have False if H: "t < y" |
|
156 |
proof - |
|
49519 | 157 |
from less_le_trans[OF lx px] less_trans[OF H yu] |
158 |
have "l < t \<and> t < u" by simp |
|
60539 | 159 |
with tU noU show ?thesis by auto |
160 |
qed |
|
49519 | 161 |
then have "\<not> t < y" by auto |
162 |
then show "y \<le> t" by (simp add: not_less) |
|
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34cb0b457dcc
Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
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163 |
qed |
34cb0b457dcc
Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff
changeset
|
164 |
|
49519 | 165 |
lemma lin_dense_ge[no_atp]: |
166 |
"t \<in> U \<Longrightarrow> |
|
60539 | 167 |
\<forall>x l u. (\<forall>t. l < t \<and> t < u \<longrightarrow> t \<notin> U) \<and> l < x \<and> x < u \<and> t \<le> x \<longrightarrow> (\<forall>y. l < y \<and> y < u \<longrightarrow> t \<le> y)" |
168 |
proof clarsimp |
|
26161
34cb0b457dcc
Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff
changeset
|
169 |
fix x l u y |
60539 | 170 |
assume tU: "t \<in> U" |
171 |
and noU: "\<forall>t. l < t \<and> t < u \<longrightarrow> t \<notin> U" |
|
172 |
and lx: "l < x" |
|
173 |
and xu: "x < u" |
|
174 |
and px: "t \<le> x" |
|
175 |
and ly: "l < y" |
|
176 |
and yu: "y < u" |
|
177 |
from tU noU ly yu have tny: "t \<noteq> y" by auto |
|
178 |
have False if H: "y < t" |
|
179 |
proof - |
|
49519 | 180 |
from less_trans[OF ly H] le_less_trans[OF px xu] |
26161
34cb0b457dcc
Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff
changeset
|
181 |
have "l < t \<and> t < u" by simp |
60539 | 182 |
with tU noU show ?thesis by auto |
183 |
qed |
|
184 |
then have "\<not> y < t" by auto |
|
49519 | 185 |
then show "t \<le> y" by (simp add: not_less) |
26161
34cb0b457dcc
Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff
changeset
|
186 |
qed |
34cb0b457dcc
Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff
changeset
|
187 |
|
49519 | 188 |
lemma lin_dense_eq[no_atp]: |
189 |
"t \<in> U \<Longrightarrow> |
|
60539 | 190 |
\<forall>x l u. (\<forall>t. l < t \<and> t < u \<longrightarrow> t \<notin> U) \<and> l < x \<and> x < u \<and> x = t \<longrightarrow> (\<forall>y. l < y \<and> y < u \<longrightarrow> y = t)" |
49519 | 191 |
by auto |
192 |
||
193 |
lemma lin_dense_neq[no_atp]: |
|
194 |
"t \<in> U \<Longrightarrow> |
|
60539 | 195 |
\<forall>x l u. (\<forall>t. l < t \<and> t < u \<longrightarrow> t \<notin> U) \<and> l < x \<and> x < u \<and> x \<noteq> t \<longrightarrow> (\<forall>y. l < y \<and> y < u \<longrightarrow> y \<noteq> t)" |
49519 | 196 |
by auto |
197 |
||
198 |
lemma lin_dense_P[no_atp]: |
|
60539 | 199 |
"\<forall>x l u. (\<forall>t. l < t \<and> t < u \<longrightarrow> t \<notin> U) \<and> l < x \<and> x < u \<and> P \<longrightarrow> (\<forall>y. l < y \<and> y < u \<longrightarrow> P)" |
49519 | 200 |
by auto |
26161
34cb0b457dcc
Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff
changeset
|
201 |
|
35828
46cfc4b8112e
now use "Named_Thms" for "noatp", and renamed "noatp" to "no_atp"
blanchet
parents:
35267
diff
changeset
|
202 |
lemma lin_dense_conj[no_atp]: |
60539 | 203 |
"\<lbrakk>\<forall>x l u. (\<forall>t. l < t \<and> t < u \<longrightarrow> t \<notin> U) \<and> l < x \<and> x < u \<and> P1 x |
204 |
\<longrightarrow> (\<forall>y. l < y \<and> y < u \<longrightarrow> P1 y) ; |
|
205 |
\<forall>x l u. (\<forall>t. l < t \<and> t < u \<longrightarrow> t \<notin> U) \<and> l < x \<and> x < u \<and> P2 x |
|
206 |
\<longrightarrow> (\<forall>y. l < y \<and> y < u \<longrightarrow> P2 y)\<rbrakk> \<Longrightarrow> |
|
207 |
\<forall>x l u. (\<forall>t. l < t \<and> t < u \<longrightarrow> t \<notin> U) \<and> l < x \<and> x < u \<and> (P1 x \<and> P2 x) |
|
208 |
\<longrightarrow> (\<forall>y. l < y \<and> y < u \<longrightarrow> (P1 y \<and> P2 y))" |
|
26161
34cb0b457dcc
Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff
changeset
|
209 |
by blast |
35828
46cfc4b8112e
now use "Named_Thms" for "noatp", and renamed "noatp" to "no_atp"
blanchet
parents:
35267
diff
changeset
|
210 |
lemma lin_dense_disj[no_atp]: |
60539 | 211 |
"\<lbrakk>\<forall>x l u. (\<forall>t. l < t \<and> t < u \<longrightarrow> t \<notin> U) \<and> l < x \<and> x < u \<and> P1 x |
212 |
\<longrightarrow> (\<forall>y. l < y \<and> y < u \<longrightarrow> P1 y) ; |
|
213 |
\<forall>x l u. (\<forall>t. l < t \<and> t < u \<longrightarrow> t \<notin> U) \<and> l < x \<and> x < u \<and> P2 x |
|
214 |
\<longrightarrow> (\<forall>y. l < y \<and> y < u \<longrightarrow> P2 y)\<rbrakk> \<Longrightarrow> |
|
215 |
\<forall>x l u. (\<forall>t. l < t \<and> t < u \<longrightarrow> t \<notin> U) \<and> l < x \<and> x < u \<and> (P1 x \<or> P2 x) |
|
216 |
\<longrightarrow> (\<forall>y. l < y \<and> y < u \<longrightarrow> (P1 y \<or> P2 y))" |
|
26161
34cb0b457dcc
Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff
changeset
|
217 |
by blast |
34cb0b457dcc
Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff
changeset
|
218 |
|
60539 | 219 |
lemma npmibnd[no_atp]: "\<lbrakk>\<forall>x. \<not> MP \<and> P x \<longrightarrow> (\<exists>u\<in> U. u \<le> x); \<forall>x. \<not>PP \<and> P x \<longrightarrow> (\<exists>u\<in> U. x \<le> u)\<rbrakk> |
220 |
\<Longrightarrow> \<forall>x. \<not> MP \<and> \<not>PP \<and> P x \<longrightarrow> (\<exists>u\<in> U. \<exists>u' \<in> U. u \<le> x \<and> x \<le> u')" |
|
49519 | 221 |
by auto |
26161
34cb0b457dcc
Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff
changeset
|
222 |
|
35828
46cfc4b8112e
now use "Named_Thms" for "noatp", and renamed "noatp" to "no_atp"
blanchet
parents:
35267
diff
changeset
|
223 |
lemma finite_set_intervals[no_atp]: |
60539 | 224 |
assumes px: "P x" |
225 |
and lx: "l \<le> x" |
|
226 |
and xu: "x \<le> u" |
|
227 |
and linS: "l\<in> S" |
|
228 |
and uinS: "u \<in> S" |
|
229 |
and fS:"finite S" |
|
230 |
and lS: "\<forall>x\<in> S. l \<le> x" |
|
231 |
and Su: "\<forall>x\<in> S. x \<le> u" |
|
232 |
shows "\<exists>a \<in> S. \<exists>b \<in> S. (\<forall>y. a < y \<and> y < b \<longrightarrow> y \<notin> S) \<and> a \<le> x \<and> x \<le> b \<and> P x" |
|
49519 | 233 |
proof - |
26161
34cb0b457dcc
Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff
changeset
|
234 |
let ?Mx = "{y. y\<in> S \<and> y \<le> x}" |
34cb0b457dcc
Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff
changeset
|
235 |
let ?xM = "{y. y\<in> S \<and> x \<le> y}" |
34cb0b457dcc
Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff
changeset
|
236 |
let ?a = "Max ?Mx" |
34cb0b457dcc
Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff
changeset
|
237 |
let ?b = "Min ?xM" |
60539 | 238 |
have MxS: "?Mx \<subseteq> S" |
239 |
by blast |
|
240 |
then have fMx: "finite ?Mx" |
|
241 |
using fS finite_subset by auto |
|
242 |
from lx linS have linMx: "l \<in> ?Mx" |
|
243 |
by blast |
|
244 |
then have Mxne: "?Mx \<noteq> {}" |
|
245 |
by blast |
|
246 |
have xMS: "?xM \<subseteq> S" |
|
247 |
by blast |
|
248 |
then have fxM: "finite ?xM" |
|
249 |
using fS finite_subset by auto |
|
250 |
from xu uinS have linxM: "u \<in> ?xM" |
|
251 |
by blast |
|
252 |
then have xMne: "?xM \<noteq> {}" |
|
253 |
by blast |
|
254 |
have ax: "?a \<le> x" |
|
255 |
using Mxne fMx by auto |
|
256 |
have xb: "x \<le> ?b" |
|
257 |
using xMne fxM by auto |
|
258 |
have "?a \<in> ?Mx" |
|
60708 | 259 |
using Max_in[OF fMx Mxne] by simp |
260 |
then have ainS: "?a \<in> S" |
|
261 |
using MxS by blast |
|
60539 | 262 |
have "?b \<in> ?xM" |
60708 | 263 |
using Min_in[OF fxM xMne] by simp |
264 |
then have binS: "?b \<in> S" |
|
265 |
using xMS by blast |
|
60539 | 266 |
have noy: "\<forall>y. ?a < y \<and> y < ?b \<longrightarrow> y \<notin> S" |
267 |
proof clarsimp |
|
49519 | 268 |
fix y |
269 |
assume ay: "?a < y" and yb: "y < ?b" and yS: "y \<in> S" |
|
60708 | 270 |
from yS have "y \<in> ?Mx \<or> y \<in> ?xM" |
271 |
by (auto simp add: linear) |
|
60539 | 272 |
then show False |
273 |
proof |
|
49519 | 274 |
assume "y \<in> ?Mx" |
60708 | 275 |
then have "y \<le> ?a" |
276 |
using Mxne fMx by auto |
|
277 |
with ay show ?thesis |
|
278 |
by (simp add: not_le[symmetric]) |
|
60539 | 279 |
next |
49519 | 280 |
assume "y \<in> ?xM" |
60708 | 281 |
then have "?b \<le> y" |
282 |
using xMne fxM by auto |
|
283 |
with yb show ?thesis |
|
284 |
by (simp add: not_le[symmetric]) |
|
60539 | 285 |
qed |
26161
34cb0b457dcc
Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff
changeset
|
286 |
qed |
60708 | 287 |
from ainS binS noy ax xb px show ?thesis |
288 |
by blast |
|
26161
34cb0b457dcc
Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff
changeset
|
289 |
qed |
34cb0b457dcc
Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff
changeset
|
290 |
|
35828
46cfc4b8112e
now use "Named_Thms" for "noatp", and renamed "noatp" to "no_atp"
blanchet
parents:
35267
diff
changeset
|
291 |
lemma finite_set_intervals2[no_atp]: |
60539 | 292 |
assumes px: "P x" |
293 |
and lx: "l \<le> x" |
|
294 |
and xu: "x \<le> u" |
|
295 |
and linS: "l\<in> S" |
|
296 |
and uinS: "u \<in> S" |
|
297 |
and fS: "finite S" |
|
298 |
and lS: "\<forall>x\<in> S. l \<le> x" |
|
299 |
and Su: "\<forall>x\<in> S. x \<le> u" |
|
300 |
shows "(\<exists>s\<in> S. P s) \<or> (\<exists>a \<in> S. \<exists>b \<in> S. (\<forall>y. a < y \<and> y < b \<longrightarrow> y \<notin> S) \<and> a < x \<and> x < b \<and> P x)" |
|
301 |
proof - |
|
26161
34cb0b457dcc
Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff
changeset
|
302 |
from finite_set_intervals[where P="P", OF px lx xu linS uinS fS lS Su] |
60539 | 303 |
obtain a and b where as: "a \<in> S" and bs: "b \<in> S" |
304 |
and noS: "\<forall>y. a < y \<and> y < b \<longrightarrow> y \<notin> S" |
|
305 |
and axb: "a \<le> x \<and> x \<le> b \<and> P x" |
|
306 |
by auto |
|
307 |
from axb have "x = a \<or> x = b \<or> (a < x \<and> x < b)" |
|
308 |
by (auto simp add: le_less) |
|
309 |
then show ?thesis |
|
310 |
using px as bs noS by blast |
|
26161
34cb0b457dcc
Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff
changeset
|
311 |
qed |
34cb0b457dcc
Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff
changeset
|
312 |
|
34cb0b457dcc
Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff
changeset
|
313 |
end |
34cb0b457dcc
Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff
changeset
|
314 |
|
49519 | 315 |
|
60533 | 316 |
section \<open>The classical QE after Langford for dense linear orders\<close> |
26161
34cb0b457dcc
Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff
changeset
|
317 |
|
53215
5e47c31c6f7c
renamed typeclass dense_linorder to unbounded_dense_linorder
hoelzl
parents:
53072
diff
changeset
|
318 |
context unbounded_dense_linorder |
26161
34cb0b457dcc
Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff
changeset
|
319 |
begin |
34cb0b457dcc
Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff
changeset
|
320 |
|
49519 | 321 |
lemma interval_empty_iff: "{y. x < y \<and> y < z} = {} \<longleftrightarrow> \<not> x < z" |
27825 | 322 |
by (auto dest: dense) |
323 |
||
60539 | 324 |
lemma dlo_qe_bnds[no_atp]: |
325 |
assumes ne: "L \<noteq> {}" |
|
326 |
and neU: "U \<noteq> {}" |
|
327 |
and fL: "finite L" |
|
328 |
and fU: "finite U" |
|
329 |
shows "(\<exists>x. (\<forall>y \<in> L. y < x) \<and> (\<forall>y \<in> U. x < y)) \<equiv> (\<forall>l \<in> L. \<forall>u \<in> U. l < u)" |
|
26161
34cb0b457dcc
Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff
changeset
|
330 |
proof (simp only: atomize_eq, rule iffI) |
34cb0b457dcc
Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff
changeset
|
331 |
assume H: "\<exists>x. (\<forall>y\<in>L. y < x) \<and> (\<forall>y\<in>U. x < y)" |
60708 | 332 |
then obtain x where xL: "\<forall>y\<in>L. y < x" and xU: "\<forall>y\<in>U. x < y" |
333 |
by blast |
|
60539 | 334 |
have "l < u" if l: "l \<in> L" and u: "u \<in> U" for l u |
335 |
proof - |
|
26161
34cb0b457dcc
Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff
changeset
|
336 |
have "l < x" using xL l by blast |
34cb0b457dcc
Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff
changeset
|
337 |
also have "x < u" using xU u by blast |
60539 | 338 |
finally show ?thesis . |
339 |
qed |
|
49519 | 340 |
then show "\<forall>l\<in>L. \<forall>u\<in>U. l < u" by blast |
26161
34cb0b457dcc
Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff
changeset
|
341 |
next |
34cb0b457dcc
Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff
changeset
|
342 |
assume H: "\<forall>l\<in>L. \<forall>u\<in>U. l < u" |
34cb0b457dcc
Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff
changeset
|
343 |
let ?ML = "Max L" |
60539 | 344 |
let ?MU = "Min U" |
60708 | 345 |
from fL ne have th1: "?ML \<in> L" and th1': "\<forall>l\<in>L. l \<le> ?ML" |
346 |
by auto |
|
347 |
from fU neU have th2: "?MU \<in> U" and th2': "\<forall>u\<in>U. ?MU \<le> u" |
|
348 |
by auto |
|
349 |
from th1 th2 H have "?ML < ?MU" |
|
350 |
by auto |
|
351 |
with dense obtain w where th3: "?ML < w" and th4: "w < ?MU" |
|
352 |
by blast |
|
353 |
from th3 th1' have "\<forall>l \<in> L. l < w" |
|
354 |
by auto |
|
355 |
moreover from th4 th2' have "\<forall>u \<in> U. w < u" |
|
356 |
by auto |
|
357 |
ultimately show "\<exists>x. (\<forall>y\<in>L. y < x) \<and> (\<forall>y\<in>U. x < y)" |
|
358 |
by auto |
|
26161
34cb0b457dcc
Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff
changeset
|
359 |
qed |
34cb0b457dcc
Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff
changeset
|
360 |
|
60539 | 361 |
lemma dlo_qe_noub[no_atp]: |
362 |
assumes ne: "L \<noteq> {}" |
|
363 |
and fL: "finite L" |
|
26161
34cb0b457dcc
Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff
changeset
|
364 |
shows "(\<exists>x. (\<forall>y \<in> L. y < x) \<and> (\<forall>y \<in> {}. x < y)) \<equiv> True" |
60539 | 365 |
proof (simp add: atomize_eq) |
60708 | 366 |
from gt_ex[of "Max L"] obtain M where M: "Max L < M" |
367 |
by blast |
|
368 |
from ne fL have "\<forall>x \<in> L. x \<le> Max L" |
|
369 |
by simp |
|
370 |
with M have "\<forall>x\<in>L. x < M" |
|
371 |
by (auto intro: le_less_trans) |
|
372 |
then show "\<exists>x. \<forall>y\<in>L. y < x" |
|
373 |
by blast |
|
26161
34cb0b457dcc
Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff
changeset
|
374 |
qed |
34cb0b457dcc
Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff
changeset
|
375 |
|
60539 | 376 |
lemma dlo_qe_nolb[no_atp]: |
377 |
assumes ne: "U \<noteq> {}" |
|
378 |
and fU: "finite U" |
|
26161
34cb0b457dcc
Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff
changeset
|
379 |
shows "(\<exists>x. (\<forall>y \<in> {}. y < x) \<and> (\<forall>y \<in> U. x < y)) \<equiv> True" |
60539 | 380 |
proof (simp add: atomize_eq) |
60708 | 381 |
from lt_ex[of "Min U"] obtain M where M: "M < Min U" |
382 |
by blast |
|
383 |
from ne fU have "\<forall>x \<in> U. Min U \<le> x" |
|
384 |
by simp |
|
385 |
with M have "\<forall>x\<in>U. M < x" |
|
386 |
by (auto intro: less_le_trans) |
|
387 |
then show "\<exists>x. \<forall>y\<in>U. x < y" |
|
388 |
by blast |
|
26161
34cb0b457dcc
Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff
changeset
|
389 |
qed |
34cb0b457dcc
Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff
changeset
|
390 |
|
60539 | 391 |
lemma exists_neq[no_atp]: "\<exists>(x::'a). x \<noteq> t" "\<exists>(x::'a). t \<noteq> x" |
26161
34cb0b457dcc
Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff
changeset
|
392 |
using gt_ex[of t] by auto |
34cb0b457dcc
Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff
changeset
|
393 |
|
60539 | 394 |
lemmas dlo_simps[no_atp] = order_refl less_irrefl not_less not_le exists_neq |
26161
34cb0b457dcc
Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff
changeset
|
395 |
le_less neq_iff linear less_not_permute |
34cb0b457dcc
Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff
changeset
|
396 |
|
67398 | 397 |
lemma axiom[no_atp]: "class.unbounded_dense_linorder (\<le>) (<)" |
53215
5e47c31c6f7c
renamed typeclass dense_linorder to unbounded_dense_linorder
hoelzl
parents:
53072
diff
changeset
|
398 |
by (rule unbounded_dense_linorder_axioms) |
35828
46cfc4b8112e
now use "Named_Thms" for "noatp", and renamed "noatp" to "no_atp"
blanchet
parents:
35267
diff
changeset
|
399 |
lemma atoms[no_atp]: |
26161
34cb0b457dcc
Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff
changeset
|
400 |
shows "TERM (less :: 'a \<Rightarrow> _)" |
34cb0b457dcc
Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff
changeset
|
401 |
and "TERM (less_eq :: 'a \<Rightarrow> _)" |
67398 | 402 |
and "TERM ((=) :: 'a \<Rightarrow> _)" . |
26161
34cb0b457dcc
Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff
changeset
|
403 |
|
34cb0b457dcc
Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff
changeset
|
404 |
declare axiom[langford qe: dlo_qe_bnds dlo_qe_nolb dlo_qe_noub gather: gather_start gather_simps atoms: atoms] |
34cb0b457dcc
Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff
changeset
|
405 |
declare dlo_simps[langfordsimp] |
34cb0b457dcc
Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff
changeset
|
406 |
|
34cb0b457dcc
Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff
changeset
|
407 |
end |
34cb0b457dcc
Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff
changeset
|
408 |
|
34cb0b457dcc
Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff
changeset
|
409 |
(* FIXME: Move to HOL -- together with the conj_aci_rule in langford.ML *) |
41849 | 410 |
lemmas dnf[no_atp] = conj_disj_distribL conj_disj_distribR |
26161
34cb0b457dcc
Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff
changeset
|
411 |
|
35828
46cfc4b8112e
now use "Named_Thms" for "noatp", and renamed "noatp" to "no_atp"
blanchet
parents:
35267
diff
changeset
|
412 |
lemmas weak_dnf_simps[no_atp] = simp_thms dnf |
26161
34cb0b457dcc
Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff
changeset
|
413 |
|
35828
46cfc4b8112e
now use "Named_Thms" for "noatp", and renamed "noatp" to "no_atp"
blanchet
parents:
35267
diff
changeset
|
414 |
lemma nnf_simps[no_atp]: |
60708 | 415 |
"(\<not> (P \<and> Q)) \<longleftrightarrow> (\<not> P \<or> \<not> Q)" |
416 |
"(\<not> (P \<or> Q)) \<longleftrightarrow> (\<not> P \<and> \<not> Q)" |
|
417 |
"(P \<longrightarrow> Q) \<longleftrightarrow> (\<not> P \<or> Q)" |
|
418 |
"(P \<longleftrightarrow> Q) \<longleftrightarrow> ((P \<and> Q) \<or> (\<not> P \<and> \<not> Q))" |
|
419 |
"(\<not> \<not> P) \<longleftrightarrow> P" |
|
26161
34cb0b457dcc
Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff
changeset
|
420 |
by blast+ |
34cb0b457dcc
Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff
changeset
|
421 |
|
60708 | 422 |
lemma ex_distrib[no_atp]: "(\<exists>x. P x \<or> Q x) \<longleftrightarrow> ((\<exists>x. P x) \<or> (\<exists>x. Q x))" |
423 |
by blast |
|
26161
34cb0b457dcc
Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff
changeset
|
424 |
|
35828
46cfc4b8112e
now use "Named_Thms" for "noatp", and renamed "noatp" to "no_atp"
blanchet
parents:
35267
diff
changeset
|
425 |
lemmas dnf_simps[no_atp] = weak_dnf_simps nnf_simps ex_distrib |
26161
34cb0b457dcc
Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff
changeset
|
426 |
|
48891 | 427 |
ML_file "langford.ML" |
60533 | 428 |
method_setup dlo = \<open> |
55848 | 429 |
Scan.succeed (SIMPLE_METHOD' o Langford.dlo_tac) |
60533 | 430 |
\<close> "Langford's algorithm for quantifier elimination in dense linear orders" |
26161
34cb0b457dcc
Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff
changeset
|
431 |
|
34cb0b457dcc
Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff
changeset
|
432 |
|
60533 | 433 |
section \<open>Contructive dense linear orders yield QE for linear arithmetic over ordered Fields\<close> |
26161
34cb0b457dcc
Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff
changeset
|
434 |
|
60533 | 435 |
text \<open>Linear order without upper bounds\<close> |
26161
34cb0b457dcc
Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff
changeset
|
436 |
|
29509
1ff0f3f08a7b
migrated class package to new locale implementation
haftmann
parents:
29252
diff
changeset
|
437 |
locale linorder_stupid_syntax = linorder |
26161
34cb0b457dcc
Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff
changeset
|
438 |
begin |
49519 | 439 |
|
26161
34cb0b457dcc
Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff
changeset
|
440 |
notation |
67398 | 441 |
less_eq ("'(\<sqsubseteq>')") and |
26161
34cb0b457dcc
Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff
changeset
|
442 |
less_eq ("(_/ \<sqsubseteq> _)" [51, 51] 50) and |
67398 | 443 |
less ("'(\<sqsubset>')") and |
26161
34cb0b457dcc
Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff
changeset
|
444 |
less ("(_/ \<sqsubset> _)" [51, 51] 50) |
34cb0b457dcc
Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff
changeset
|
445 |
|
34cb0b457dcc
Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff
changeset
|
446 |
end |
34cb0b457dcc
Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff
changeset
|
447 |
|
29509
1ff0f3f08a7b
migrated class package to new locale implementation
haftmann
parents:
29252
diff
changeset
|
448 |
locale linorder_no_ub = linorder_stupid_syntax + |
26161
34cb0b457dcc
Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff
changeset
|
449 |
assumes gt_ex: "\<exists>y. less x y" |
34cb0b457dcc
Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff
changeset
|
450 |
begin |
49519 | 451 |
|
60539 | 452 |
lemma ge_ex[no_atp]: "\<exists>y. x \<sqsubseteq> y" |
453 |
using gt_ex by auto |
|
26161
34cb0b457dcc
Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff
changeset
|
454 |
|
61586 | 455 |
text \<open>Theorems for \<open>\<exists>z. \<forall>x. z \<sqsubset> x \<longrightarrow> (P x \<longleftrightarrow> P\<^sub>+\<^sub>\<infinity>)\<close>\<close> |
35828
46cfc4b8112e
now use "Named_Thms" for "noatp", and renamed "noatp" to "no_atp"
blanchet
parents:
35267
diff
changeset
|
456 |
lemma pinf_conj[no_atp]: |
26161
34cb0b457dcc
Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff
changeset
|
457 |
assumes ex1: "\<exists>z1. \<forall>x. z1 \<sqsubset> x \<longrightarrow> (P1 x \<longleftrightarrow> P1')" |
60539 | 458 |
and ex2: "\<exists>z2. \<forall>x. z2 \<sqsubset> x \<longrightarrow> (P2 x \<longleftrightarrow> P2')" |
26161
34cb0b457dcc
Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff
changeset
|
459 |
shows "\<exists>z. \<forall>x. z \<sqsubset> x \<longrightarrow> ((P1 x \<and> P2 x) \<longleftrightarrow> (P1' \<and> P2'))" |
60539 | 460 |
proof - |
60708 | 461 |
from ex1 ex2 obtain z1 and z2 |
462 |
where z1: "\<forall>x. z1 \<sqsubset> x \<longrightarrow> (P1 x \<longleftrightarrow> P1')" |
|
463 |
and z2: "\<forall>x. z2 \<sqsubset> x \<longrightarrow> (P2 x \<longleftrightarrow> P2')" |
|
464 |
by blast |
|
465 |
from gt_ex obtain z where z:"ord.max less_eq z1 z2 \<sqsubset> z" |
|
466 |
by blast |
|
467 |
from z have zz1: "z1 \<sqsubset> z" and zz2: "z2 \<sqsubset> z" |
|
468 |
by simp_all |
|
60539 | 469 |
have "(P1 x \<and> P2 x) \<longleftrightarrow> (P1' \<and> P2')" if H: "z \<sqsubset> x" for x |
470 |
using less_trans[OF zz1 H] less_trans[OF zz2 H] z1 zz1 z2 zz2 by auto |
|
60708 | 471 |
then show ?thesis |
472 |
by blast |
|
26161
34cb0b457dcc
Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff
changeset
|
473 |
qed |
34cb0b457dcc
Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff
changeset
|
474 |
|
35828
46cfc4b8112e
now use "Named_Thms" for "noatp", and renamed "noatp" to "no_atp"
blanchet
parents:
35267
diff
changeset
|
475 |
lemma pinf_disj[no_atp]: |
26161
34cb0b457dcc
Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff
changeset
|
476 |
assumes ex1: "\<exists>z1. \<forall>x. z1 \<sqsubset> x \<longrightarrow> (P1 x \<longleftrightarrow> P1')" |
49519 | 477 |
and ex2: "\<exists>z2. \<forall>x. z2 \<sqsubset> x \<longrightarrow> (P2 x \<longleftrightarrow> P2')" |
26161
34cb0b457dcc
Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff
changeset
|
478 |
shows "\<exists>z. \<forall>x. z \<sqsubset> x \<longrightarrow> ((P1 x \<or> P2 x) \<longleftrightarrow> (P1' \<or> P2'))" |
34cb0b457dcc
Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff
changeset
|
479 |
proof- |
60708 | 480 |
from ex1 ex2 obtain z1 and z2 |
481 |
where z1: "\<forall>x. z1 \<sqsubset> x \<longrightarrow> (P1 x \<longleftrightarrow> P1')" |
|
482 |
and z2: "\<forall>x. z2 \<sqsubset> x \<longrightarrow> (P2 x \<longleftrightarrow> P2')" |
|
483 |
by blast |
|
484 |
from gt_ex obtain z where z: "ord.max less_eq z1 z2 \<sqsubset> z" |
|
485 |
by blast |
|
486 |
from z have zz1: "z1 \<sqsubset> z" and zz2: "z2 \<sqsubset> z" |
|
487 |
by simp_all |
|
60539 | 488 |
have "(P1 x \<or> P2 x) \<longleftrightarrow> (P1' \<or> P2')" if H: "z \<sqsubset> x" for x |
489 |
using less_trans[OF zz1 H] less_trans[OF zz2 H] z1 zz1 z2 zz2 by auto |
|
60708 | 490 |
then show ?thesis |
491 |
by blast |
|
26161
34cb0b457dcc
Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff
changeset
|
492 |
qed |
34cb0b457dcc
Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff
changeset
|
493 |
|
60539 | 494 |
lemma pinf_ex[no_atp]: |
495 |
assumes ex: "\<exists>z. \<forall>x. z \<sqsubset> x \<longrightarrow> (P x \<longleftrightarrow> P1)" |
|
496 |
and p1: P1 |
|
497 |
shows "\<exists>x. P x" |
|
49519 | 498 |
proof - |
60708 | 499 |
from ex obtain z where z: "\<forall>x. z \<sqsubset> x \<longrightarrow> (P x \<longleftrightarrow> P1)" |
500 |
by blast |
|
501 |
from gt_ex obtain x where x: "z \<sqsubset> x" |
|
502 |
by blast |
|
503 |
from z x p1 show ?thesis |
|
504 |
by blast |
|
26161
34cb0b457dcc
Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff
changeset
|
505 |
qed |
34cb0b457dcc
Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff
changeset
|
506 |
|
34cb0b457dcc
Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff
changeset
|
507 |
end |
34cb0b457dcc
Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff
changeset
|
508 |
|
60533 | 509 |
text \<open>Linear order without upper bounds\<close> |
26161
34cb0b457dcc
Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff
changeset
|
510 |
|
29509
1ff0f3f08a7b
migrated class package to new locale implementation
haftmann
parents:
29252
diff
changeset
|
511 |
locale linorder_no_lb = linorder_stupid_syntax + |
26161
34cb0b457dcc
Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff
changeset
|
512 |
assumes lt_ex: "\<exists>y. less y x" |
34cb0b457dcc
Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff
changeset
|
513 |
begin |
49519 | 514 |
|
60708 | 515 |
lemma le_ex[no_atp]: "\<exists>y. y \<sqsubseteq> x" |
516 |
using lt_ex by auto |
|
26161
34cb0b457dcc
Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff
changeset
|
517 |
|
34cb0b457dcc
Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff
changeset
|
518 |
|
61586 | 519 |
text \<open>Theorems for \<open>\<exists>z. \<forall>x. x \<sqsubset> z \<longrightarrow> (P x \<longleftrightarrow> P\<^sub>-\<^sub>\<infinity>)\<close>\<close> |
35828
46cfc4b8112e
now use "Named_Thms" for "noatp", and renamed "noatp" to "no_atp"
blanchet
parents:
35267
diff
changeset
|
520 |
lemma minf_conj[no_atp]: |
26161
34cb0b457dcc
Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff
changeset
|
521 |
assumes ex1: "\<exists>z1. \<forall>x. x \<sqsubset> z1 \<longrightarrow> (P1 x \<longleftrightarrow> P1')" |
49519 | 522 |
and ex2: "\<exists>z2. \<forall>x. x \<sqsubset> z2 \<longrightarrow> (P2 x \<longleftrightarrow> P2')" |
26161
34cb0b457dcc
Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff
changeset
|
523 |
shows "\<exists>z. \<forall>x. x \<sqsubset> z \<longrightarrow> ((P1 x \<and> P2 x) \<longleftrightarrow> (P1' \<and> P2'))" |
60539 | 524 |
proof - |
60708 | 525 |
from ex1 ex2 obtain z1 and z2 |
526 |
where z1: "\<forall>x. x \<sqsubset> z1 \<longrightarrow> (P1 x \<longleftrightarrow> P1')" |
|
60539 | 527 |
and z2: "\<forall>x. x \<sqsubset> z2 \<longrightarrow> (P2 x \<longleftrightarrow> P2')" |
528 |
by blast |
|
60708 | 529 |
from lt_ex obtain z where z: "z \<sqsubset> ord.min less_eq z1 z2" |
530 |
by blast |
|
531 |
from z have zz1: "z \<sqsubset> z1" and zz2: "z \<sqsubset> z2" |
|
532 |
by simp_all |
|
60539 | 533 |
have "(P1 x \<and> P2 x) \<longleftrightarrow> (P1' \<and> P2')" if H: "x \<sqsubset> z" for x |
534 |
using less_trans[OF H zz1] less_trans[OF H zz2] z1 zz1 z2 zz2 by auto |
|
60708 | 535 |
then show ?thesis |
536 |
by blast |
|
26161
34cb0b457dcc
Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff
changeset
|
537 |
qed |
34cb0b457dcc
Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff
changeset
|
538 |
|
35828
46cfc4b8112e
now use "Named_Thms" for "noatp", and renamed "noatp" to "no_atp"
blanchet
parents:
35267
diff
changeset
|
539 |
lemma minf_disj[no_atp]: |
26161
34cb0b457dcc
Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff
changeset
|
540 |
assumes ex1: "\<exists>z1. \<forall>x. x \<sqsubset> z1 \<longrightarrow> (P1 x \<longleftrightarrow> P1')" |
49519 | 541 |
and ex2: "\<exists>z2. \<forall>x. x \<sqsubset> z2 \<longrightarrow> (P2 x \<longleftrightarrow> P2')" |
26161
34cb0b457dcc
Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff
changeset
|
542 |
shows "\<exists>z. \<forall>x. x \<sqsubset> z \<longrightarrow> ((P1 x \<or> P2 x) \<longleftrightarrow> (P1' \<or> P2'))" |
49519 | 543 |
proof - |
60708 | 544 |
from ex1 ex2 obtain z1 and z2 |
545 |
where z1: "\<forall>x. x \<sqsubset> z1 \<longrightarrow> (P1 x \<longleftrightarrow> P1')" |
|
546 |
and z2: "\<forall>x. x \<sqsubset> z2 \<longrightarrow> (P2 x \<longleftrightarrow> P2')" |
|
547 |
by blast |
|
548 |
from lt_ex obtain z where z: "z \<sqsubset> ord.min less_eq z1 z2" |
|
549 |
by blast |
|
550 |
from z have zz1: "z \<sqsubset> z1" and zz2: "z \<sqsubset> z2" |
|
551 |
by simp_all |
|
60539 | 552 |
have "(P1 x \<or> P2 x) \<longleftrightarrow> (P1' \<or> P2')" if H: "x \<sqsubset> z" for x |
553 |
using less_trans[OF H zz1] less_trans[OF H zz2] z1 zz1 z2 zz2 by auto |
|
60708 | 554 |
then show ?thesis |
555 |
by blast |
|
26161
34cb0b457dcc
Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff
changeset
|
556 |
qed |
34cb0b457dcc
Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff
changeset
|
557 |
|
49519 | 558 |
lemma minf_ex[no_atp]: |
559 |
assumes ex: "\<exists>z. \<forall>x. x \<sqsubset> z \<longrightarrow> (P x \<longleftrightarrow> P1)" |
|
560 |
and p1: P1 |
|
60539 | 561 |
shows "\<exists>x. P x" |
49519 | 562 |
proof - |
60708 | 563 |
from ex obtain z where z: "\<forall>x. x \<sqsubset> z \<longrightarrow> (P x \<longleftrightarrow> P1)" |
564 |
by blast |
|
565 |
from lt_ex obtain x where x: "x \<sqsubset> z" |
|
566 |
by blast |
|
567 |
from z x p1 show ?thesis |
|
568 |
by blast |
|
26161
34cb0b457dcc
Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff
changeset
|
569 |
qed |
34cb0b457dcc
Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff
changeset
|
570 |
|
34cb0b457dcc
Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff
changeset
|
571 |
end |
34cb0b457dcc
Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff
changeset
|
572 |
|
34cb0b457dcc
Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff
changeset
|
573 |
|
35028
108662d50512
more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents:
34974
diff
changeset
|
574 |
locale constr_dense_linorder = linorder_no_lb + linorder_no_ub + |
26161
34cb0b457dcc
Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff
changeset
|
575 |
fixes between |
34cb0b457dcc
Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff
changeset
|
576 |
assumes between_less: "less x y \<Longrightarrow> less x (between x y) \<and> less (between x y) y" |
49519 | 577 |
and between_same: "between x x = x" |
54868 | 578 |
begin |
26161
34cb0b457dcc
Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff
changeset
|
579 |
|
60539 | 580 |
sublocale dlo: unbounded_dense_linorder |
61166
5976fe402824
renamed method "goals" to "goal_cases" to emphasize its meaning;
wenzelm
parents:
61089
diff
changeset
|
581 |
proof (unfold_locales, goal_cases) |
60708 | 582 |
case (1 x y) |
583 |
then show ?case |
|
584 |
using between_less [of x y] by auto |
|
585 |
next |
|
586 |
case 2 |
|
587 |
then show ?case by (rule lt_ex) |
|
588 |
next |
|
589 |
case 3 |
|
590 |
then show ?case by (rule gt_ex) |
|
591 |
qed |
|
26161
34cb0b457dcc
Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff
changeset
|
592 |
|
35828
46cfc4b8112e
now use "Named_Thms" for "noatp", and renamed "noatp" to "no_atp"
blanchet
parents:
35267
diff
changeset
|
593 |
lemma rinf_U[no_atp]: |
26161
34cb0b457dcc
Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff
changeset
|
594 |
assumes fU: "finite U" |
60539 | 595 |
and lin_dense: "\<forall>x l u. (\<forall>t. l \<sqsubset> t \<and> t\<sqsubset> u \<longrightarrow> t \<notin> U) \<and> l\<sqsubset> x \<and> x \<sqsubset> u \<and> P x |
596 |
\<longrightarrow> (\<forall>y. l \<sqsubset> y \<and> y \<sqsubset> u \<longrightarrow> P y )" |
|
597 |
and nmpiU: "\<forall>x. \<not> MP \<and> \<not>PP \<and> P x \<longrightarrow> (\<exists>u\<in> U. \<exists>u' \<in> U. u \<sqsubseteq> x \<and> x \<sqsubseteq> u')" |
|
598 |
and nmi: "\<not> MP" and npi: "\<not> PP" and ex: "\<exists>x. P x" |
|
599 |
shows "\<exists>u\<in> U. \<exists>u' \<in> U. P (between u u')" |
|
49519 | 600 |
proof - |
60539 | 601 |
from ex obtain x where px: "P x" |
602 |
by blast |
|
603 |
from px nmi npi nmpiU have "\<exists>u\<in> U. \<exists>u' \<in> U. u \<sqsubseteq> x \<and> x \<sqsubseteq> u'" |
|
604 |
by auto |
|
605 |
then obtain u and u' where uU: "u\<in> U" and uU': "u' \<in> U" and ux: "u \<sqsubseteq> x" and xu': "x \<sqsubseteq> u'" |
|
606 |
by auto |
|
607 |
from uU have Une: "U \<noteq> {}" |
|
608 |
by auto |
|
26161
34cb0b457dcc
Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff
changeset
|
609 |
let ?l = "linorder.Min less_eq U" |
34cb0b457dcc
Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff
changeset
|
610 |
let ?u = "linorder.Max less_eq U" |
60539 | 611 |
have linM: "?l \<in> U" |
612 |
using fU Une by simp |
|
613 |
have uinM: "?u \<in> U" |
|
614 |
using fU Une by simp |
|
615 |
have lM: "\<forall>t\<in> U. ?l \<sqsubseteq> t" |
|
616 |
using Une fU by auto |
|
617 |
have Mu: "\<forall>t\<in> U. t \<sqsubseteq> ?u" |
|
618 |
using Une fU by auto |
|
619 |
have th: "?l \<sqsubseteq> u" |
|
620 |
using uU Une lM by auto |
|
26161
34cb0b457dcc
Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff
changeset
|
621 |
from order_trans[OF th ux] have lx: "?l \<sqsubseteq> x" . |
60539 | 622 |
have th: "u' \<sqsubseteq> ?u" |
623 |
using uU' Une Mu by simp |
|
26161
34cb0b457dcc
Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff
changeset
|
624 |
from order_trans[OF xu' th] have xu: "x \<sqsubseteq> ?u" . |
34cb0b457dcc
Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff
changeset
|
625 |
from finite_set_intervals2[where P="P",OF px lx xu linM uinM fU lM Mu] |
60539 | 626 |
consider u where "u \<in> U" "P u" | |
627 |
t1 t2 where "t1 \<in> U" "t2 \<in> U" "\<forall>y. t1 \<sqsubset> y \<and> y \<sqsubset> t2 \<longrightarrow> y \<notin> U" "t1 \<sqsubset> x" "x \<sqsubset> t2" "P x" |
|
628 |
by blast |
|
629 |
then show ?thesis |
|
630 |
proof cases |
|
60567 | 631 |
case u: 1 |
26161
34cb0b457dcc
Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff
changeset
|
632 |
have "between u u = u" by (simp add: between_same) |
60567 | 633 |
with u have "P (between u u)" by simp |
634 |
with u show ?thesis by blast |
|
60539 | 635 |
next |
636 |
case 2 |
|
637 |
note t1M = \<open>t1 \<in> U\<close> and t2M = \<open>t2\<in> U\<close> |
|
638 |
and noM = \<open>\<forall>y. t1 \<sqsubset> y \<and> y \<sqsubset> t2 \<longrightarrow> y \<notin> U\<close> |
|
639 |
and t1x = \<open>t1 \<sqsubset> x\<close> and xt2 = \<open>x \<sqsubset> t2\<close> |
|
640 |
and px = \<open>P x\<close> |
|
49519 | 641 |
from less_trans[OF t1x xt2] have t1t2: "t1 \<sqsubset> t2" . |
642 |
let ?u = "between t1 t2" |
|
643 |
from between_less t1t2 have t1lu: "t1 \<sqsubset> ?u" and ut2: "?u \<sqsubset> t2" by auto |
|
644 |
from lin_dense noM t1x xt2 px t1lu ut2 have "P ?u" by blast |
|
60539 | 645 |
with t1M t2M show ?thesis by blast |
646 |
qed |
|
49519 | 647 |
qed |
26161
34cb0b457dcc
Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff
changeset
|
648 |
|
35828
46cfc4b8112e
now use "Named_Thms" for "noatp", and renamed "noatp" to "no_atp"
blanchet
parents:
35267
diff
changeset
|
649 |
theorem fr_eq[no_atp]: |
26161
34cb0b457dcc
Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff
changeset
|
650 |
assumes fU: "finite U" |
60539 | 651 |
and lin_dense: "\<forall>x l u. (\<forall>t. l \<sqsubset> t \<and> t\<sqsubset> u \<longrightarrow> t \<notin> U) \<and> l\<sqsubset> x \<and> x \<sqsubset> u \<and> P x |
652 |
\<longrightarrow> (\<forall>y. l \<sqsubset> y \<and> y \<sqsubset> u \<longrightarrow> P y )" |
|
653 |
and nmibnd: "\<forall>x. \<not> MP \<and> P x \<longrightarrow> (\<exists>u\<in> U. u \<sqsubseteq> x)" |
|
654 |
and npibnd: "\<forall>x. \<not>PP \<and> P x \<longrightarrow> (\<exists>u\<in> U. x \<sqsubseteq> u)" |
|
49519 | 655 |
and mi: "\<exists>z. \<forall>x. x \<sqsubset> z \<longrightarrow> (P x = MP)" and pi: "\<exists>z. \<forall>x. z \<sqsubset> x \<longrightarrow> (P x = PP)" |
60539 | 656 |
shows "(\<exists>x. P x) \<equiv> (MP \<or> PP \<or> (\<exists>u \<in> U. \<exists>u'\<in> U. P (between u u')))" |
26161
34cb0b457dcc
Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff
changeset
|
657 |
(is "_ \<equiv> (_ \<or> _ \<or> ?F)" is "?E \<equiv> ?D") |
49519 | 658 |
proof - |
60539 | 659 |
have "?E \<longleftrightarrow> ?D" |
660 |
proof |
|
661 |
show ?D if px: ?E |
|
662 |
proof - |
|
663 |
consider "MP \<or> PP" | "\<not> MP" "\<not> PP" by blast |
|
664 |
then show ?thesis |
|
665 |
proof cases |
|
666 |
case 1 |
|
667 |
then show ?thesis by blast |
|
668 |
next |
|
669 |
case 2 |
|
670 |
from npmibnd[OF nmibnd npibnd] |
|
671 |
have nmpiU: "\<forall>x. \<not> MP \<and> \<not>PP \<and> P x \<longrightarrow> (\<exists>u\<in> U. \<exists>u' \<in> U. u \<sqsubseteq> x \<and> x \<sqsubseteq> u')" . |
|
672 |
from rinf_U[OF fU lin_dense nmpiU \<open>\<not> MP\<close> \<open>\<not> PP\<close> px] show ?thesis |
|
673 |
by blast |
|
674 |
qed |
|
675 |
qed |
|
676 |
show ?E if ?D |
|
677 |
proof - |
|
678 |
from that consider MP | PP | ?F by blast |
|
679 |
then show ?thesis |
|
680 |
proof cases |
|
681 |
case 1 |
|
60567 | 682 |
from minf_ex[OF mi this] show ?thesis . |
60539 | 683 |
next |
684 |
case 2 |
|
60567 | 685 |
from pinf_ex[OF pi this] show ?thesis . |
60539 | 686 |
next |
687 |
case 3 |
|
688 |
then show ?thesis by blast |
|
689 |
qed |
|
690 |
qed |
|
691 |
qed |
|
692 |
then show "?E \<equiv> ?D" by simp |
|
26161
34cb0b457dcc
Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff
changeset
|
693 |
qed |
34cb0b457dcc
Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff
changeset
|
694 |
|
35828
46cfc4b8112e
now use "Named_Thms" for "noatp", and renamed "noatp" to "no_atp"
blanchet
parents:
35267
diff
changeset
|
695 |
lemmas minf_thms[no_atp] = minf_conj minf_disj minf_eq minf_neq minf_lt minf_le minf_gt minf_ge minf_P |
46cfc4b8112e
now use "Named_Thms" for "noatp", and renamed "noatp" to "no_atp"
blanchet
parents:
35267
diff
changeset
|
696 |
lemmas pinf_thms[no_atp] = pinf_conj pinf_disj pinf_eq pinf_neq pinf_lt pinf_le pinf_gt pinf_ge pinf_P |
26161
34cb0b457dcc
Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff
changeset
|
697 |
|
35828
46cfc4b8112e
now use "Named_Thms" for "noatp", and renamed "noatp" to "no_atp"
blanchet
parents:
35267
diff
changeset
|
698 |
lemmas nmi_thms[no_atp] = nmi_conj nmi_disj nmi_eq nmi_neq nmi_lt nmi_le nmi_gt nmi_ge nmi_P |
46cfc4b8112e
now use "Named_Thms" for "noatp", and renamed "noatp" to "no_atp"
blanchet
parents:
35267
diff
changeset
|
699 |
lemmas npi_thms[no_atp] = npi_conj npi_disj npi_eq npi_neq npi_lt npi_le npi_gt npi_ge npi_P |
46cfc4b8112e
now use "Named_Thms" for "noatp", and renamed "noatp" to "no_atp"
blanchet
parents:
35267
diff
changeset
|
700 |
lemmas lin_dense_thms[no_atp] = lin_dense_conj lin_dense_disj lin_dense_eq lin_dense_neq lin_dense_lt lin_dense_le lin_dense_gt lin_dense_ge lin_dense_P |
26161
34cb0b457dcc
Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff
changeset
|
701 |
|
35828
46cfc4b8112e
now use "Named_Thms" for "noatp", and renamed "noatp" to "no_atp"
blanchet
parents:
35267
diff
changeset
|
702 |
lemma ferrack_axiom[no_atp]: "constr_dense_linorder less_eq less between" |
35028
108662d50512
more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents:
34974
diff
changeset
|
703 |
by (rule constr_dense_linorder_axioms) |
49519 | 704 |
|
35828
46cfc4b8112e
now use "Named_Thms" for "noatp", and renamed "noatp" to "no_atp"
blanchet
parents:
35267
diff
changeset
|
705 |
lemma atoms[no_atp]: |
26161
34cb0b457dcc
Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff
changeset
|
706 |
shows "TERM (less :: 'a \<Rightarrow> _)" |
34cb0b457dcc
Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff
changeset
|
707 |
and "TERM (less_eq :: 'a \<Rightarrow> _)" |
67398 | 708 |
and "TERM ((=) :: 'a \<Rightarrow> _)" . |
26161
34cb0b457dcc
Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff
changeset
|
709 |
|
34cb0b457dcc
Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff
changeset
|
710 |
declare ferrack_axiom [ferrack minf: minf_thms pinf: pinf_thms |
34cb0b457dcc
Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff
changeset
|
711 |
nmi: nmi_thms npi: npi_thms lindense: |
34cb0b457dcc
Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff
changeset
|
712 |
lin_dense_thms qe: fr_eq atoms: atoms] |
34cb0b457dcc
Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff
changeset
|
713 |
|
60533 | 714 |
declaration \<open> |
26161
34cb0b457dcc
Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff
changeset
|
715 |
let |
49519 | 716 |
fun simps phi = map (Morphism.thm phi) [@{thm "not_less"}, @{thm "not_le"}] |
717 |
fun generic_whatis phi = |
|
718 |
let |
|
69597 | 719 |
val [lt, le] = map (Morphism.term phi) [\<^term>\<open>(\<sqsubset>)\<close>, \<^term>\<open>(\<sqsubseteq>)\<close>] |
49519 | 720 |
fun h x t = |
59582 | 721 |
case Thm.term_of t of |
69597 | 722 |
Const(\<^const_name>\<open>HOL.eq\<close>, _)$y$z => |
59582 | 723 |
if Thm.term_of x aconv y then Ferrante_Rackoff_Data.Eq |
49519 | 724 |
else Ferrante_Rackoff_Data.Nox |
69597 | 725 |
| \<^term>\<open>Not\<close>$(Const(\<^const_name>\<open>HOL.eq\<close>, _)$y$z) => |
59582 | 726 |
if Thm.term_of x aconv y then Ferrante_Rackoff_Data.NEq |
49519 | 727 |
else Ferrante_Rackoff_Data.Nox |
728 |
| b$y$z => if Term.could_unify (b, lt) then |
|
59582 | 729 |
if Thm.term_of x aconv y then Ferrante_Rackoff_Data.Lt |
730 |
else if Thm.term_of x aconv z then Ferrante_Rackoff_Data.Gt |
|
49519 | 731 |
else Ferrante_Rackoff_Data.Nox |
732 |
else if Term.could_unify (b, le) then |
|
59582 | 733 |
if Thm.term_of x aconv y then Ferrante_Rackoff_Data.Le |
734 |
else if Thm.term_of x aconv z then Ferrante_Rackoff_Data.Ge |
|
49519 | 735 |
else Ferrante_Rackoff_Data.Nox |
26161
34cb0b457dcc
Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff
changeset
|
736 |
else Ferrante_Rackoff_Data.Nox |
49519 | 737 |
| _ => Ferrante_Rackoff_Data.Nox |
738 |
in h end |
|
61089 | 739 |
fun ss phi ctxt = |
740 |
simpset_of (put_simpset HOL_ss ctxt addsimps (simps phi)) |
|
26161
34cb0b457dcc
Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff
changeset
|
741 |
in |
49519 | 742 |
Ferrante_Rackoff_Data.funs @{thm "ferrack_axiom"} |
743 |
{isolate_conv = K (K (K Thm.reflexive)), whatis = generic_whatis, simpset = ss} |
|
26161
34cb0b457dcc
Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff
changeset
|
744 |
end |
60533 | 745 |
\<close> |
26161
34cb0b457dcc
Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff
changeset
|
746 |
|
34cb0b457dcc
Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff
changeset
|
747 |
end |
34cb0b457dcc
Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff
changeset
|
748 |
|
48891 | 749 |
ML_file "ferrante_rackoff.ML" |
26161
34cb0b457dcc
Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff
changeset
|
750 |
|
60533 | 751 |
method_setup ferrack = \<open> |
30549 | 752 |
Scan.succeed (SIMPLE_METHOD' o FerranteRackoff.dlo_tac) |
60533 | 753 |
\<close> "Ferrante and Rackoff's algorithm for quantifier elimination in dense linear orders" |
26161
34cb0b457dcc
Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff
changeset
|
754 |
|
49519 | 755 |
|
60533 | 756 |
subsection \<open>Ferrante and Rackoff algorithm over ordered fields\<close> |
26161
34cb0b457dcc
Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff
changeset
|
757 |
|
60539 | 758 |
lemma neg_prod_lt: |
759 |
fixes c :: "'a::linordered_field" |
|
760 |
assumes "c < 0" |
|
761 |
shows "c * x < 0 \<equiv> x > 0" |
|
49519 | 762 |
proof - |
60539 | 763 |
have "c * x < 0 \<longleftrightarrow> 0 / c < x" |
764 |
by (simp only: neg_divide_less_eq[OF \<open>c < 0\<close>] algebra_simps) |
|
765 |
also have "\<dots> \<longleftrightarrow> 0 < x" by simp |
|
766 |
finally show "PROP ?thesis" by simp |
|
26161
34cb0b457dcc
Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff
changeset
|
767 |
qed |
34cb0b457dcc
Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff
changeset
|
768 |
|
60539 | 769 |
lemma pos_prod_lt: |
770 |
fixes c :: "'a::linordered_field" |
|
771 |
assumes "c > 0" |
|
772 |
shows "c * x < 0 \<equiv> x < 0" |
|
49519 | 773 |
proof - |
60539 | 774 |
have "c * x < 0 \<longleftrightarrow> 0 /c > x" |
775 |
by (simp only: pos_less_divide_eq[OF \<open>c > 0\<close>] algebra_simps) |
|
776 |
also have "\<dots> \<longleftrightarrow> 0 > x" by simp |
|
777 |
finally show "PROP ?thesis" by simp |
|
26161
34cb0b457dcc
Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff
changeset
|
778 |
qed |
34cb0b457dcc
Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff
changeset
|
779 |
|
60539 | 780 |
lemma neg_prod_sum_lt: |
781 |
fixes c :: "'a::linordered_field" |
|
782 |
assumes "c < 0" |
|
783 |
shows "c * x + t < 0 \<equiv> x > (- 1 / c) * t" |
|
49519 | 784 |
proof - |
60539 | 785 |
have "c * x + t < 0 \<longleftrightarrow> c * x < - t" |
786 |
by (subst less_iff_diff_less_0 [of "c * x" "- t"]) simp |
|
787 |
also have "\<dots> \<longleftrightarrow> - t / c < x" |
|
788 |
by (simp only: neg_divide_less_eq[OF \<open>c < 0\<close>] algebra_simps) |
|
789 |
also have "\<dots> \<longleftrightarrow> (- 1 / c) * t < x" by simp |
|
790 |
finally show "PROP ?thesis" by simp |
|
26161
34cb0b457dcc
Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff
changeset
|
791 |
qed |
34cb0b457dcc
Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff
changeset
|
792 |
|
60539 | 793 |
lemma pos_prod_sum_lt: |
794 |
fixes c :: "'a::linordered_field" |
|
795 |
assumes "c > 0" |
|
796 |
shows "c * x + t < 0 \<equiv> x < (- 1 / c) * t" |
|
49519 | 797 |
proof - |
60539 | 798 |
have "c * x + t < 0 \<longleftrightarrow> c * x < - t" |
799 |
by (subst less_iff_diff_less_0 [of "c * x" "- t"]) simp |
|
800 |
also have "\<dots> \<longleftrightarrow> - t / c > x" |
|
801 |
by (simp only: pos_less_divide_eq[OF \<open>c > 0\<close>] algebra_simps) |
|
802 |
also have "\<dots> \<longleftrightarrow> (- 1 / c) * t > x" by simp |
|
803 |
finally show "PROP ?thesis" by simp |
|
26161
34cb0b457dcc
Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff
changeset
|
804 |
qed |
34cb0b457dcc
Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff
changeset
|
805 |
|
60539 | 806 |
lemma sum_lt: |
807 |
fixes x :: "'a::ordered_ab_group_add" |
|
808 |
shows "x + t < 0 \<equiv> x < - t" |
|
26161
34cb0b457dcc
Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff
changeset
|
809 |
using less_diff_eq[where a= x and b=t and c=0] by simp |
34cb0b457dcc
Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff
changeset
|
810 |
|
60539 | 811 |
lemma neg_prod_le: |
812 |
fixes c :: "'a::linordered_field" |
|
813 |
assumes "c < 0" |
|
814 |
shows "c * x \<le> 0 \<equiv> x \<ge> 0" |
|
49519 | 815 |
proof - |
60539 | 816 |
have "c * x \<le> 0 \<longleftrightarrow> 0 / c \<le> x" |
817 |
by (simp only: neg_divide_le_eq[OF \<open>c < 0\<close>] algebra_simps) |
|
818 |
also have "\<dots> \<longleftrightarrow> 0 \<le> x" by simp |
|
819 |
finally show "PROP ?thesis" by simp |
|
26161
34cb0b457dcc
Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff
changeset
|
820 |
qed |
34cb0b457dcc
Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff
changeset
|
821 |
|
60539 | 822 |
lemma pos_prod_le: |
823 |
fixes c :: "'a::linordered_field" |
|
824 |
assumes "c > 0" |
|
825 |
shows "c * x \<le> 0 \<equiv> x \<le> 0" |
|
49519 | 826 |
proof - |
60539 | 827 |
have "c * x \<le> 0 \<longleftrightarrow> 0 / c \<ge> x" |
828 |
by (simp only: pos_le_divide_eq[OF \<open>c > 0\<close>] algebra_simps) |
|
829 |
also have "\<dots> \<longleftrightarrow> 0 \<ge> x" by simp |
|
830 |
finally show "PROP ?thesis" by simp |
|
831 |
qed |
|
832 |
||
833 |
lemma neg_prod_sum_le: |
|
834 |
fixes c :: "'a::linordered_field" |
|
835 |
assumes "c < 0" |
|
836 |
shows "c * x + t \<le> 0 \<equiv> x \<ge> (- 1 / c) * t" |
|
837 |
proof - |
|
838 |
have "c * x + t \<le> 0 \<longleftrightarrow> c * x \<le> -t" |
|
839 |
by (subst le_iff_diff_le_0 [of "c*x" "-t"]) simp |
|
840 |
also have "\<dots> \<longleftrightarrow> - t / c \<le> x" |
|
841 |
by (simp only: neg_divide_le_eq[OF \<open>c < 0\<close>] algebra_simps) |
|
842 |
also have "\<dots> \<longleftrightarrow> (- 1 / c) * t \<le> x" by simp |
|
843 |
finally show "PROP ?thesis" by simp |
|
26161
34cb0b457dcc
Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff
changeset
|
844 |
qed |
34cb0b457dcc
Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff
changeset
|
845 |
|
60539 | 846 |
lemma pos_prod_sum_le: |
847 |
fixes c :: "'a::linordered_field" |
|
848 |
assumes "c > 0" |
|
849 |
shows "c * x + t \<le> 0 \<equiv> x \<le> (- 1 / c) * t" |
|
49519 | 850 |
proof - |
60539 | 851 |
have "c * x + t \<le> 0 \<longleftrightarrow> c * x \<le> - t" |
852 |
by (subst le_iff_diff_le_0 [of "c*x" "-t"]) simp |
|
853 |
also have "\<dots> \<longleftrightarrow> - t / c \<ge> x" |
|
854 |
by (simp only: pos_le_divide_eq[OF \<open>c > 0\<close>] algebra_simps) |
|
855 |
also have "\<dots> \<longleftrightarrow> (- 1 / c) * t \<ge> x" by simp |
|
856 |
finally show "PROP ?thesis" by simp |
|
26161
34cb0b457dcc
Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff
changeset
|
857 |
qed |
34cb0b457dcc
Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff
changeset
|
858 |
|
60539 | 859 |
lemma sum_le: |
860 |
fixes x :: "'a::ordered_ab_group_add" |
|
861 |
shows "x + t \<le> 0 \<equiv> x \<le> - t" |
|
26161
34cb0b457dcc
Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff
changeset
|
862 |
using le_diff_eq[where a= x and b=t and c=0] by simp |
34cb0b457dcc
Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff
changeset
|
863 |
|
60539 | 864 |
lemma nz_prod_eq: |
865 |
fixes c :: "'a::linordered_field" |
|
866 |
assumes "c \<noteq> 0" |
|
867 |
shows "c * x = 0 \<equiv> x = 0" |
|
868 |
using assms by simp |
|
49519 | 869 |
|
60539 | 870 |
lemma nz_prod_sum_eq: |
871 |
fixes c :: "'a::linordered_field" |
|
872 |
assumes "c \<noteq> 0" |
|
873 |
shows "c * x + t = 0 \<equiv> x = (- 1/c) * t" |
|
49519 | 874 |
proof - |
60539 | 875 |
have "c * x + t = 0 \<longleftrightarrow> c * x = - t" |
876 |
by (subst eq_iff_diff_eq_0 [of "c*x" "-t"]) simp |
|
877 |
also have "\<dots> \<longleftrightarrow> x = - t / c" |
|
878 |
by (simp only: nonzero_eq_divide_eq[OF \<open>c \<noteq> 0\<close>] algebra_simps) |
|
879 |
finally show "PROP ?thesis" by simp |
|
26161
34cb0b457dcc
Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff
changeset
|
880 |
qed |
49519 | 881 |
|
60539 | 882 |
lemma sum_eq: |
883 |
fixes x :: "'a::ordered_ab_group_add" |
|
884 |
shows "x + t = 0 \<equiv> x = - t" |
|
26161
34cb0b457dcc
Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff
changeset
|
885 |
using eq_diff_eq[where a= x and b=t and c=0] by simp |
34cb0b457dcc
Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff
changeset
|
886 |
|
35043
07dbdf60d5ad
dropped accidental duplication of "lin" prefix from cs. 108662d50512
haftmann
parents:
35028
diff
changeset
|
887 |
interpretation class_dense_linordered_field: constr_dense_linorder |
67398 | 888 |
"(\<le>)" "(<)" "\<lambda>x y. 1/2 * ((x::'a::linordered_field) + y)" |
49519 | 889 |
by unfold_locales (dlo, dlo, auto) |
26161
34cb0b457dcc
Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff
changeset
|
890 |
|
60539 | 891 |
declaration \<open> |
26161
34cb0b457dcc
Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff
changeset
|
892 |
let |
67562 | 893 |
fun earlier [] _ = false |
894 |
| earlier (h::t) (x, y) = |
|
895 |
if h aconvc y then false else if h aconvc x then true else earlier t (x, y); |
|
896 |
||
67563 | 897 |
fun earlier_ord vs (x, y) = |
67565 | 898 |
if x aconvc y then EQUAL |
899 |
else if earlier vs (x, y) then LESS |
|
900 |
else GREATER; |
|
26161
34cb0b457dcc
Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff
changeset
|
901 |
|
49519 | 902 |
fun dest_frac ct = |
59582 | 903 |
case Thm.term_of ct of |
69597 | 904 |
Const (\<^const_name>\<open>Rings.divide\<close>,_) $ a $ b=> |
63201 | 905 |
Rat.make (snd (HOLogic.dest_number a), snd (HOLogic.dest_number b)) |
69597 | 906 |
| Const(\<^const_name>\<open>inverse\<close>, _)$a => Rat.make(1, HOLogic.dest_number a |> snd) |
63201 | 907 |
| t => Rat.of_int (snd (HOLogic.dest_number t)) |
26161
34cb0b457dcc
Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff
changeset
|
908 |
|
59582 | 909 |
fun whatis x ct = case Thm.term_of ct of |
69597 | 910 |
Const(\<^const_name>\<open>Groups.plus\<close>, _)$(Const(\<^const_name>\<open>Groups.times\<close>,_)$_$y)$_ => |
59582 | 911 |
if y aconv Thm.term_of x then ("c*x+t",[(funpow 2 Thm.dest_arg1) ct, Thm.dest_arg ct]) |
26161
34cb0b457dcc
Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff
changeset
|
912 |
else ("Nox",[]) |
69597 | 913 |
| Const(\<^const_name>\<open>Groups.plus\<close>, _)$y$_ => |
59582 | 914 |
if y aconv Thm.term_of x then ("x+t",[Thm.dest_arg ct]) |
26161
34cb0b457dcc
Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff
changeset
|
915 |
else ("Nox",[]) |
69597 | 916 |
| Const(\<^const_name>\<open>Groups.times\<close>, _)$_$y => |
59582 | 917 |
if y aconv Thm.term_of x then ("c*x",[Thm.dest_arg1 ct]) |
26161
34cb0b457dcc
Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff
changeset
|
918 |
else ("Nox",[]) |
59582 | 919 |
| t => if t aconv Thm.term_of x then ("x",[]) else ("Nox",[]); |
26161
34cb0b457dcc
Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff
changeset
|
920 |
|
36945 | 921 |
fun xnormalize_conv ctxt [] ct = Thm.reflexive ct |
26161
34cb0b457dcc
Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff
changeset
|
922 |
| xnormalize_conv ctxt (vs as (x::_)) ct = |
59582 | 923 |
case Thm.term_of ct of |
69597 | 924 |
Const(\<^const_name>\<open>Orderings.less\<close>,_)$_$Const(\<^const_name>\<open>Groups.zero\<close>,_) => |
26161
34cb0b457dcc
Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff
changeset
|
925 |
(case whatis x (Thm.dest_arg1 ct) of |
34cb0b457dcc
Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff
changeset
|
926 |
("c*x+t",[c,t]) => |
34cb0b457dcc
Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff
changeset
|
927 |
let |
34cb0b457dcc
Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff
changeset
|
928 |
val cr = dest_frac c |
34cb0b457dcc
Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff
changeset
|
929 |
val clt = Thm.dest_fun2 ct |
34cb0b457dcc
Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff
changeset
|
930 |
val cz = Thm.dest_arg ct |
63205 | 931 |
val neg = cr < @0 |
51717
9e7d1c139569
simplifier uses proper Proof.context instead of historic type simpset;
wenzelm
parents:
49756
diff
changeset
|
932 |
val cthp = Simplifier.rewrite ctxt |
69597 | 933 |
(Thm.apply \<^cterm>\<open>Trueprop\<close> |
46497
89ccf66aa73d
renamed Thm.capply to Thm.apply, and Thm.cabs to Thm.lambda in conformance with similar operations in structure Term and Logic;
wenzelm
parents:
45620
diff
changeset
|
934 |
(if neg then Thm.apply (Thm.apply clt c) cz |
89ccf66aa73d
renamed Thm.capply to Thm.apply, and Thm.cabs to Thm.lambda in conformance with similar operations in structure Term and Logic;
wenzelm
parents:
45620
diff
changeset
|
935 |
else Thm.apply (Thm.apply clt cz) c)) |
36945 | 936 |
val cth = Thm.equal_elim (Thm.symmetric cthp) TrueI |
60801 | 937 |
val th = Thm.implies_elim (Thm.instantiate' [SOME (Thm.ctyp_of_cterm x)] (map SOME [c,x,t]) |
26161
34cb0b457dcc
Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff
changeset
|
938 |
(if neg then @{thm neg_prod_sum_lt} else @{thm pos_prod_sum_lt})) cth |
34cb0b457dcc
Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff
changeset
|
939 |
val rth = Conv.fconv_rule (Conv.arg_conv (Conv.binop_conv |
67562 | 940 |
(Semiring_Normalizer.semiring_normalize_ord_conv ctxt (earlier_ord vs)))) th |
26161
34cb0b457dcc
Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff
changeset
|
941 |
in rth end |
34cb0b457dcc
Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff
changeset
|
942 |
| ("x+t",[t]) => |
34cb0b457dcc
Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff
changeset
|
943 |
let |
59586 | 944 |
val T = Thm.ctyp_of_cterm x |
60801 | 945 |
val th = Thm.instantiate' [SOME T] [SOME x, SOME t] @{thm "sum_lt"} |
26161
34cb0b457dcc
Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff
changeset
|
946 |
val rth = Conv.fconv_rule (Conv.arg_conv (Conv.binop_conv |
67562 | 947 |
(Semiring_Normalizer.semiring_normalize_ord_conv ctxt (earlier_ord vs)))) th |
26161
34cb0b457dcc
Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff
changeset
|
948 |
in rth end |
34cb0b457dcc
Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff
changeset
|
949 |
| ("c*x",[c]) => |
34cb0b457dcc
Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff
changeset
|
950 |
let |
34cb0b457dcc
Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff
changeset
|
951 |
val cr = dest_frac c |
34cb0b457dcc
Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff
changeset
|
952 |
val clt = Thm.dest_fun2 ct |
34cb0b457dcc
Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff
changeset
|
953 |
val cz = Thm.dest_arg ct |
63205 | 954 |
val neg = cr < @0 |
51717
9e7d1c139569
simplifier uses proper Proof.context instead of historic type simpset;
wenzelm
parents:
49756
diff
changeset
|
955 |
val cthp = Simplifier.rewrite ctxt |
69597 | 956 |
(Thm.apply \<^cterm>\<open>Trueprop\<close> |
46497
89ccf66aa73d
renamed Thm.capply to Thm.apply, and Thm.cabs to Thm.lambda in conformance with similar operations in structure Term and Logic;
wenzelm
parents:
45620
diff
changeset
|
957 |
(if neg then Thm.apply (Thm.apply clt c) cz |
89ccf66aa73d
renamed Thm.capply to Thm.apply, and Thm.cabs to Thm.lambda in conformance with similar operations in structure Term and Logic;
wenzelm
parents:
45620
diff
changeset
|
958 |
else Thm.apply (Thm.apply clt cz) c)) |
36945 | 959 |
val cth = Thm.equal_elim (Thm.symmetric cthp) TrueI |
60801 | 960 |
val th = Thm.implies_elim (Thm.instantiate' [SOME (Thm.ctyp_of_cterm x)] (map SOME [c,x]) |
26161
34cb0b457dcc
Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff
changeset
|
961 |
(if neg then @{thm neg_prod_lt} else @{thm pos_prod_lt})) cth |
34cb0b457dcc
Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff
changeset
|
962 |
val rth = th |
34cb0b457dcc
Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff
changeset
|
963 |
in rth end |
36945 | 964 |
| _ => Thm.reflexive ct) |
26161
34cb0b457dcc
Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff
changeset
|
965 |
|
34cb0b457dcc
Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff
changeset
|
966 |
|
69597 | 967 |
| Const(\<^const_name>\<open>Orderings.less_eq\<close>,_)$_$Const(\<^const_name>\<open>Groups.zero\<close>,_) => |
26161
34cb0b457dcc
Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff
changeset
|
968 |
(case whatis x (Thm.dest_arg1 ct) of |
34cb0b457dcc
Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff
changeset
|
969 |
("c*x+t",[c,t]) => |
34cb0b457dcc
Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff
changeset
|
970 |
let |
61075 | 971 |
val T = Thm.typ_of_cterm x |
972 |
val cT = Thm.ctyp_of_cterm x |
|
26161
34cb0b457dcc
Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff
changeset
|
973 |
val cr = dest_frac c |
69597 | 974 |
val clt = Thm.cterm_of ctxt (Const (\<^const_name>\<open>ord_class.less\<close>, T --> T --> \<^typ>\<open>bool\<close>)) |
26161
34cb0b457dcc
Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff
changeset
|
975 |
val cz = Thm.dest_arg ct |
63205 | 976 |
val neg = cr < @0 |
51717
9e7d1c139569
simplifier uses proper Proof.context instead of historic type simpset;
wenzelm
parents:
49756
diff
changeset
|
977 |
val cthp = Simplifier.rewrite ctxt |
69597 | 978 |
(Thm.apply \<^cterm>\<open>Trueprop\<close> |
46497
89ccf66aa73d
renamed Thm.capply to Thm.apply, and Thm.cabs to Thm.lambda in conformance with similar operations in structure Term and Logic;
wenzelm
parents:
45620
diff
changeset
|
979 |
(if neg then Thm.apply (Thm.apply clt c) cz |
89ccf66aa73d
renamed Thm.capply to Thm.apply, and Thm.cabs to Thm.lambda in conformance with similar operations in structure Term and Logic;
wenzelm
parents:
45620
diff
changeset
|
980 |
else Thm.apply (Thm.apply clt cz) c)) |
36945 | 981 |
val cth = Thm.equal_elim (Thm.symmetric cthp) TrueI |
61075 | 982 |
val th = Thm.implies_elim (Thm.instantiate' [SOME cT] (map SOME [c,x,t]) |
26161
34cb0b457dcc
Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff
changeset
|
983 |
(if neg then @{thm neg_prod_sum_le} else @{thm pos_prod_sum_le})) cth |
34cb0b457dcc
Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff
changeset
|
984 |
val rth = Conv.fconv_rule (Conv.arg_conv (Conv.binop_conv |
67562 | 985 |
(Semiring_Normalizer.semiring_normalize_ord_conv ctxt (earlier_ord vs)))) th |
26161
34cb0b457dcc
Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff
changeset
|
986 |
in rth end |
34cb0b457dcc
Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff
changeset
|
987 |
| ("x+t",[t]) => |
34cb0b457dcc
Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff
changeset
|
988 |
let |
59586 | 989 |
val T = Thm.ctyp_of_cterm x |
60801 | 990 |
val th = Thm.instantiate' [SOME T] [SOME x, SOME t] @{thm "sum_le"} |
26161
34cb0b457dcc
Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff
changeset
|
991 |
val rth = Conv.fconv_rule (Conv.arg_conv (Conv.binop_conv |
67562 | 992 |
(Semiring_Normalizer.semiring_normalize_ord_conv ctxt (earlier_ord vs)))) th |
26161
34cb0b457dcc
Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff
changeset
|
993 |
in rth end |
34cb0b457dcc
Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff
changeset
|
994 |
| ("c*x",[c]) => |
34cb0b457dcc
Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff
changeset
|
995 |
let |
61075 | 996 |
val T = Thm.typ_of_cterm x |
997 |
val cT = Thm.ctyp_of_cterm x |
|
26161
34cb0b457dcc
Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff
changeset
|
998 |
val cr = dest_frac c |
69597 | 999 |
val clt = Thm.cterm_of ctxt (Const (\<^const_name>\<open>ord_class.less\<close>, T --> T --> \<^typ>\<open>bool\<close>)) |
26161
34cb0b457dcc
Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff
changeset
|
1000 |
val cz = Thm.dest_arg ct |
63205 | 1001 |
val neg = cr < @0 |
51717
9e7d1c139569
simplifier uses proper Proof.context instead of historic type simpset;
wenzelm
parents:
49756
diff
changeset
|
1002 |
val cthp = Simplifier.rewrite ctxt |
69597 | 1003 |
(Thm.apply \<^cterm>\<open>Trueprop\<close> |
46497
89ccf66aa73d
renamed Thm.capply to Thm.apply, and Thm.cabs to Thm.lambda in conformance with similar operations in structure Term and Logic;
wenzelm
parents:
45620
diff
changeset
|
1004 |
(if neg then Thm.apply (Thm.apply clt c) cz |
89ccf66aa73d
renamed Thm.capply to Thm.apply, and Thm.cabs to Thm.lambda in conformance with similar operations in structure Term and Logic;
wenzelm
parents:
45620
diff
changeset
|
1005 |
else Thm.apply (Thm.apply clt cz) c)) |
36945 | 1006 |
val cth = Thm.equal_elim (Thm.symmetric cthp) TrueI |
60801 | 1007 |
val th = Thm.implies_elim (Thm.instantiate' [SOME (Thm.ctyp_of_cterm x)] (map SOME [c,x]) |
26161
34cb0b457dcc
Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff
changeset
|
1008 |
(if neg then @{thm neg_prod_le} else @{thm pos_prod_le})) cth |
34cb0b457dcc
Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff
changeset
|
1009 |
val rth = th |
34cb0b457dcc
Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff
changeset
|
1010 |
in rth end |
36945 | 1011 |
| _ => Thm.reflexive ct) |
26161
34cb0b457dcc
Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff
changeset
|
1012 |
|
69597 | 1013 |
| Const(\<^const_name>\<open>HOL.eq\<close>,_)$_$Const(\<^const_name>\<open>Groups.zero\<close>,_) => |
26161
34cb0b457dcc
Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff
changeset
|
1014 |
(case whatis x (Thm.dest_arg1 ct) of |
34cb0b457dcc
Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff
changeset
|
1015 |
("c*x+t",[c,t]) => |
34cb0b457dcc
Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff
changeset
|
1016 |
let |
59586 | 1017 |
val T = Thm.ctyp_of_cterm x |
26161
34cb0b457dcc
Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff
changeset
|
1018 |
val cr = dest_frac c |
34cb0b457dcc
Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff
changeset
|
1019 |
val ceq = Thm.dest_fun2 ct |
34cb0b457dcc
Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff
changeset
|
1020 |
val cz = Thm.dest_arg ct |
51717
9e7d1c139569
simplifier uses proper Proof.context instead of historic type simpset;
wenzelm
parents:
49756
diff
changeset
|
1021 |
val cthp = Simplifier.rewrite ctxt |
69597 | 1022 |
(Thm.apply \<^cterm>\<open>Trueprop\<close> |
1023 |
(Thm.apply \<^cterm>\<open>Not\<close> (Thm.apply (Thm.apply ceq c) cz))) |
|
36945 | 1024 |
val cth = Thm.equal_elim (Thm.symmetric cthp) TrueI |
1025 |
val th = Thm.implies_elim |
|
60801 | 1026 |
(Thm.instantiate' [SOME T] (map SOME [c,x,t]) @{thm nz_prod_sum_eq}) cth |
26161
34cb0b457dcc
Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff
changeset
|
1027 |
val rth = Conv.fconv_rule (Conv.arg_conv (Conv.binop_conv |
67562 | 1028 |
(Semiring_Normalizer.semiring_normalize_ord_conv ctxt (earlier_ord vs)))) th |
26161
34cb0b457dcc
Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff
changeset
|
1029 |
in rth end |
34cb0b457dcc
Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff
changeset
|
1030 |
| ("x+t",[t]) => |
34cb0b457dcc
Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff
changeset
|
1031 |
let |
59586 | 1032 |
val T = Thm.ctyp_of_cterm x |
60801 | 1033 |
val th = Thm.instantiate' [SOME T] [SOME x, SOME t] @{thm "sum_eq"} |
26161
34cb0b457dcc
Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff
changeset
|
1034 |
val rth = Conv.fconv_rule (Conv.arg_conv (Conv.binop_conv |
67562 | 1035 |
(Semiring_Normalizer.semiring_normalize_ord_conv ctxt (earlier_ord vs)))) th |
26161
34cb0b457dcc
Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff
changeset
|
1036 |
in rth end |
34cb0b457dcc
Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff
changeset
|
1037 |
| ("c*x",[c]) => |
34cb0b457dcc
Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff
changeset
|
1038 |
let |
59586 | 1039 |
val T = Thm.ctyp_of_cterm x |
26161
34cb0b457dcc
Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff
changeset
|
1040 |
val cr = dest_frac c |
34cb0b457dcc
Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff
changeset
|
1041 |
val ceq = Thm.dest_fun2 ct |
34cb0b457dcc
Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff
changeset
|
1042 |
val cz = Thm.dest_arg ct |
51717
9e7d1c139569
simplifier uses proper Proof.context instead of historic type simpset;
wenzelm
parents:
49756
diff
changeset
|
1043 |
val cthp = Simplifier.rewrite ctxt |
69597 | 1044 |
(Thm.apply \<^cterm>\<open>Trueprop\<close> |
1045 |
(Thm.apply \<^cterm>\<open>Not\<close> (Thm.apply (Thm.apply ceq c) cz))) |
|
36945 | 1046 |
val cth = Thm.equal_elim (Thm.symmetric cthp) TrueI |
1047 |
val rth = Thm.implies_elim |
|
60801 | 1048 |
(Thm.instantiate' [SOME T] (map SOME [c,x]) @{thm nz_prod_eq}) cth |
26161
34cb0b457dcc
Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff
changeset
|
1049 |
in rth end |
36945 | 1050 |
| _ => Thm.reflexive ct); |
26161
34cb0b457dcc
Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff
changeset
|
1051 |
|
34cb0b457dcc
Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff
changeset
|
1052 |
local |
34cb0b457dcc
Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff
changeset
|
1053 |
val less_iff_diff_less_0 = mk_meta_eq @{thm "less_iff_diff_less_0"} |
34cb0b457dcc
Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff
changeset
|
1054 |
val le_iff_diff_le_0 = mk_meta_eq @{thm "le_iff_diff_le_0"} |
34cb0b457dcc
Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff
changeset
|
1055 |
val eq_iff_diff_eq_0 = mk_meta_eq @{thm "eq_iff_diff_eq_0"} |
69597 | 1056 |
val ss = simpset_of \<^context> |
26161
34cb0b457dcc
Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff
changeset
|
1057 |
in |
59582 | 1058 |
fun field_isolate_conv phi ctxt vs ct = case Thm.term_of ct of |
69597 | 1059 |
Const(\<^const_name>\<open>Orderings.less\<close>,_)$a$b => |
26161
34cb0b457dcc
Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff
changeset
|
1060 |
let val (ca,cb) = Thm.dest_binop ct |
59586 | 1061 |
val T = Thm.ctyp_of_cterm ca |
60801 | 1062 |
val th = Thm.instantiate' [SOME T] [SOME ca, SOME cb] less_iff_diff_less_0 |
26161
34cb0b457dcc
Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff
changeset
|
1063 |
val nth = Conv.fconv_rule |
34cb0b457dcc
Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff
changeset
|
1064 |
(Conv.arg_conv (Conv.arg1_conv |
67562 | 1065 |
(Semiring_Normalizer.semiring_normalize_ord_conv (put_simpset ss ctxt) (earlier_ord vs)))) th |
36945 | 1066 |
val rth = Thm.transitive nth (xnormalize_conv ctxt vs (Thm.rhs_of nth)) |
26161
34cb0b457dcc
Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff
changeset
|
1067 |
in rth end |
69597 | 1068 |
| Const(\<^const_name>\<open>Orderings.less_eq\<close>,_)$a$b => |
26161
34cb0b457dcc
Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff
changeset
|
1069 |
let val (ca,cb) = Thm.dest_binop ct |
59586 | 1070 |
val T = Thm.ctyp_of_cterm ca |
60801 | 1071 |
val th = Thm.instantiate' [SOME T] [SOME ca, SOME cb] le_iff_diff_le_0 |
26161
34cb0b457dcc
Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff
changeset
|
1072 |
val nth = Conv.fconv_rule |
34cb0b457dcc
Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff
changeset
|
1073 |
(Conv.arg_conv (Conv.arg1_conv |
67562 | 1074 |
(Semiring_Normalizer.semiring_normalize_ord_conv (put_simpset ss ctxt) (earlier_ord vs)))) th |
36945 | 1075 |
val rth = Thm.transitive nth (xnormalize_conv ctxt vs (Thm.rhs_of nth)) |
26161
34cb0b457dcc
Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff
changeset
|
1076 |
in rth end |
34cb0b457dcc
Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff
changeset
|
1077 |
|
69597 | 1078 |
| Const(\<^const_name>\<open>HOL.eq\<close>,_)$a$b => |
26161
34cb0b457dcc
Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff
changeset
|
1079 |
let val (ca,cb) = Thm.dest_binop ct |
59586 | 1080 |
val T = Thm.ctyp_of_cterm ca |
60801 | 1081 |
val th = Thm.instantiate' [SOME T] [SOME ca, SOME cb] eq_iff_diff_eq_0 |
26161
34cb0b457dcc
Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff
changeset
|
1082 |
val nth = Conv.fconv_rule |
34cb0b457dcc
Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff
changeset
|
1083 |
(Conv.arg_conv (Conv.arg1_conv |
67562 | 1084 |
(Semiring_Normalizer.semiring_normalize_ord_conv (put_simpset ss ctxt) (earlier_ord vs)))) th |
36945 | 1085 |
val rth = Thm.transitive nth (xnormalize_conv ctxt vs (Thm.rhs_of nth)) |
26161
34cb0b457dcc
Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff
changeset
|
1086 |
in rth end |
69597 | 1087 |
| \<^term>\<open>Not\<close> $(Const(\<^const_name>\<open>HOL.eq\<close>,_)$a$b) => Conv.arg_conv (field_isolate_conv phi ctxt vs) ct |
36945 | 1088 |
| _ => Thm.reflexive ct |
26161
34cb0b457dcc
Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff
changeset
|
1089 |
end; |
34cb0b457dcc
Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff
changeset
|
1090 |
|
34cb0b457dcc
Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff
changeset
|
1091 |
fun classfield_whatis phi = |
34cb0b457dcc
Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff
changeset
|
1092 |
let |
34cb0b457dcc
Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff
changeset
|
1093 |
fun h x t = |
59582 | 1094 |
case Thm.term_of t of |
69597 | 1095 |
Const(\<^const_name>\<open>HOL.eq\<close>, _)$y$z => |
59582 | 1096 |
if Thm.term_of x aconv y then Ferrante_Rackoff_Data.Eq |
1097 |
else Ferrante_Rackoff_Data.Nox |
|
69597 | 1098 |
| \<^term>\<open>Not\<close>$(Const(\<^const_name>\<open>HOL.eq\<close>, _)$y$z) => |
59582 | 1099 |
if Thm.term_of x aconv y then Ferrante_Rackoff_Data.NEq |
1100 |
else Ferrante_Rackoff_Data.Nox |
|
69597 | 1101 |
| Const(\<^const_name>\<open>Orderings.less\<close>,_)$y$z => |
59582 | 1102 |
if Thm.term_of x aconv y then Ferrante_Rackoff_Data.Lt |
1103 |
else if Thm.term_of x aconv z then Ferrante_Rackoff_Data.Gt |
|
1104 |
else Ferrante_Rackoff_Data.Nox |
|
69597 | 1105 |
| Const (\<^const_name>\<open>Orderings.less_eq\<close>,_)$y$z => |
59582 | 1106 |
if Thm.term_of x aconv y then Ferrante_Rackoff_Data.Le |
1107 |
else if Thm.term_of x aconv z then Ferrante_Rackoff_Data.Ge |
|
1108 |
else Ferrante_Rackoff_Data.Nox |
|
26161
34cb0b457dcc
Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff
changeset
|
1109 |
| _ => Ferrante_Rackoff_Data.Nox |
34cb0b457dcc
Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff
changeset
|
1110 |
in h end; |
61089 | 1111 |
fun class_field_ss phi ctxt = |
1112 |
simpset_of (put_simpset HOL_basic_ss ctxt |
|
51717
9e7d1c139569
simplifier uses proper Proof.context instead of historic type simpset;
wenzelm
parents:
49756
diff
changeset
|
1113 |
addsimps ([@{thm "linorder_not_less"}, @{thm "linorder_not_le"}]) |
9e7d1c139569
simplifier uses proper Proof.context instead of historic type simpset;
wenzelm
parents:
49756
diff
changeset
|
1114 |
|> fold Splitter.add_split [@{thm "abs_split"}, @{thm "split_max"}, @{thm "split_min"}]) |
26161
34cb0b457dcc
Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff
changeset
|
1115 |
|
34cb0b457dcc
Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff
changeset
|
1116 |
in |
35043
07dbdf60d5ad
dropped accidental duplication of "lin" prefix from cs. 108662d50512
haftmann
parents:
35028
diff
changeset
|
1117 |
Ferrante_Rackoff_Data.funs @{thm "class_dense_linordered_field.ferrack_axiom"} |
26161
34cb0b457dcc
Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff
changeset
|
1118 |
{isolate_conv = field_isolate_conv, whatis = classfield_whatis, simpset = class_field_ss} |
34cb0b457dcc
Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff
changeset
|
1119 |
end |
60533 | 1120 |
\<close> |
29841
86d94bb79226
A generic decision procedure for linear rea arithmetic and normed vector spaces
chaieb
parents:
29823
diff
changeset
|
1121 |
|
60539 | 1122 |
end |